| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 7-s + 8-s + 9-s + 10-s − 11-s − 12-s − 13-s − 14-s − 15-s + 16-s + 17-s + 18-s − 19-s + 20-s + 21-s − 22-s + 23-s − 24-s + 25-s − 26-s − 27-s − 28-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 7-s + 8-s + 9-s + 10-s − 11-s − 12-s − 13-s − 14-s − 15-s + 16-s + 17-s + 18-s − 19-s + 20-s + 21-s − 22-s + 23-s − 24-s + 25-s − 26-s − 27-s − 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 631 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 631 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.127302348\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.127302348\) |
| \(L(1)\) |
\(\approx\) |
\(1.625842774\) |
| \(L(1)\) |
\(\approx\) |
\(1.625842774\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 631 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 + T \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−22.87365772070175556778551084186, −21.93529760885686915589934925048, −21.34174872673127338880339923704, −20.784593047733220160024707084929, −19.35146585910739051247907166795, −18.766708206373426251259509619209, −17.38208650643486769956848021799, −16.97134396591526479730190820659, −16.03875323999791534563669397474, −15.339909449241916412605389024285, −14.22320222706348072641814419295, −13.32554525753380880684914008123, −12.56496541679654886514215158334, −12.22840083144502334959372867798, −10.68172016498179100807754309824, −10.37995525098026001984383888753, −9.43337071381404016261148443494, −7.65843824997991087341238807444, −6.71995670949513943076103968203, −6.06406608681739157690291022903, −5.267034593544418441600269624211, −4.56040741156009783488823739622, −3.08491006324326659680591562107, −2.25609249844910983310915838122, −0.81215409254204030728460761450,
0.81215409254204030728460761450, 2.25609249844910983310915838122, 3.08491006324326659680591562107, 4.56040741156009783488823739622, 5.267034593544418441600269624211, 6.06406608681739157690291022903, 6.71995670949513943076103968203, 7.65843824997991087341238807444, 9.43337071381404016261148443494, 10.37995525098026001984383888753, 10.68172016498179100807754309824, 12.22840083144502334959372867798, 12.56496541679654886514215158334, 13.32554525753380880684914008123, 14.22320222706348072641814419295, 15.339909449241916412605389024285, 16.03875323999791534563669397474, 16.97134396591526479730190820659, 17.38208650643486769956848021799, 18.766708206373426251259509619209, 19.35146585910739051247907166795, 20.784593047733220160024707084929, 21.34174872673127338880339923704, 21.93529760885686915589934925048, 22.87365772070175556778551084186
