L(s) = 1 | + (1.73 + 2.23i)2-s + (−3.46 − 3.87i)3-s + (−2.00 + 7.74i)4-s − 8.94i·5-s + (2.66 − 14.4i)6-s + 7.74i·7-s + (−20.7 + 8.94i)8-s + (−3.00 + 26.8i)9-s + (20.0 − 15.4i)10-s + 34.6·11-s + (36.9 − 19.0i)12-s − 10·13-s + (−17.3 + 13.4i)14-s + (−34.6 + 30.9i)15-s + (−56 − 30.9i)16-s − 35.7i·17-s + ⋯ |
L(s) = 1 | + (0.612 + 0.790i)2-s + (−0.666 − 0.745i)3-s + (−0.250 + 0.968i)4-s − 0.799i·5-s + (0.181 − 0.983i)6-s + 0.418i·7-s + (−0.918 + 0.395i)8-s + (−0.111 + 0.993i)9-s + (0.632 − 0.489i)10-s + 0.949·11-s + (0.888 − 0.459i)12-s − 0.213·13-s + (−0.330 + 0.256i)14-s + (−0.596 + 0.533i)15-s + (−0.875 − 0.484i)16-s − 0.510i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.459i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.888 - 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.972332 + 0.236425i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.972332 + 0.236425i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.73 - 2.23i)T \) |
| 3 | \( 1 + (3.46 + 3.87i)T \) |
good | 5 | \( 1 + 8.94iT - 125T^{2} \) |
| 7 | \( 1 - 7.74iT - 343T^{2} \) |
| 11 | \( 1 - 34.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 10T + 2.19e3T^{2} \) |
| 17 | \( 1 + 35.7iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 69.7iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 96.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 152. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 224. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 130T + 5.06e4T^{2} \) |
| 41 | \( 1 - 125. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 224. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 193.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 545. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 173.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 442T + 2.26e5T^{2} \) |
| 67 | \( 1 - 735. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 1.03e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 410T + 3.89e5T^{2} \) |
| 79 | \( 1 + 85.2iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 1.25e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 840. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 770T + 9.12e5T^{2} \) |
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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
−19.91408288706256799828994702466, −18.12779904782448996444243581822, −17.02781762442769073291185344769, −15.98956461369380405702956087309, −14.14962300175953305361265734246, −12.74846519852760496323923348560, −11.78638494336544608530849049629, −8.710033085418608479691832702699, −6.84183802607566256049597967972, −5.11970921045757825472644203448,
3.96721015998628628573876114770, 6.19761641252313747646880668290, 9.765570252606697516390225853530, 10.90848303310514202961708056084, 12.13622797696963170452285217551, 14.14424290558261059727321999050, 15.21922772648573504637430320642, 17.01298372103877452068597504567, 18.52855858576108479790220107579, 19.96227817988236175426714493795