| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s − 11-s − 12-s + 16-s − 17-s − 19-s − 22-s − 24-s + 27-s + 32-s + 33-s − 34-s − 38-s − 41-s + 2·43-s − 44-s − 48-s + 49-s + 51-s + 54-s + 57-s + 2·59-s + 64-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s − 11-s − 12-s + 16-s − 17-s − 19-s − 22-s − 24-s + 27-s + 32-s + 33-s − 34-s − 38-s − 41-s + 2·43-s − 44-s − 48-s + 49-s + 51-s + 54-s + 57-s + 2·59-s + 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8774526908\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8774526908\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + T + T^{2} \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( ( 1 - T )^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 + T + T^{2} \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( ( 1 - T )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.71453521232408321490887288002, −11.79047453166468326613272993011, −10.94259595160588286575068372387, −10.34856892141264561174209855532, −8.562388167989146598500816226073, −7.21128182081198107883185711576, −6.18472913252883355576995074341, −5.32800143650698036361882960749, −4.28007171288186451228709174476, −2.52234865889530633241732947678,
2.52234865889530633241732947678, 4.28007171288186451228709174476, 5.32800143650698036361882960749, 6.18472913252883355576995074341, 7.21128182081198107883185711576, 8.562388167989146598500816226073, 10.34856892141264561174209855532, 10.94259595160588286575068372387, 11.79047453166468326613272993011, 12.71453521232408321490887288002
