| L(s) = 1 | + 5-s − 4·7-s + 11-s − 13-s − 3·17-s + 7·19-s − 2·23-s + 25-s + 8·29-s + 2·31-s − 4·35-s + 7·37-s + 3·41-s − 9·43-s + 11·47-s + 9·49-s − 12·53-s + 55-s − 10·59-s − 6·61-s − 65-s + 9·67-s + 2·71-s − 4·73-s − 4·77-s + 14·79-s − 6·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.51·7-s + 0.301·11-s − 0.277·13-s − 0.727·17-s + 1.60·19-s − 0.417·23-s + 1/5·25-s + 1.48·29-s + 0.359·31-s − 0.676·35-s + 1.15·37-s + 0.468·41-s − 1.37·43-s + 1.60·47-s + 9/7·49-s − 1.64·53-s + 0.134·55-s − 1.30·59-s − 0.768·61-s − 0.124·65-s + 1.09·67-s + 0.237·71-s − 0.468·73-s − 0.455·77-s + 1.57·79-s − 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.036479494\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.036479494\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−13.71340527979579, −13.31463678213126, −12.74375460755683, −12.26623200713094, −11.94409046108491, −11.28383669678605, −10.72119298751649, −10.12127089708159, −9.749435875862523, −9.364958555917556, −8.999364388223489, −8.230422268701525, −7.696369555242142, −7.078481694137870, −6.564069856720376, −6.198732430899215, −5.747953606980431, −4.947677947972161, −4.509528617795354, −3.757559484242269, −3.078050210787061, −2.823168799196777, −2.042244236686485, −1.145659166761945, −0.4859991466794404,
0.4859991466794404, 1.145659166761945, 2.042244236686485, 2.823168799196777, 3.078050210787061, 3.757559484242269, 4.509528617795354, 4.947677947972161, 5.747953606980431, 6.198732430899215, 6.564069856720376, 7.078481694137870, 7.696369555242142, 8.230422268701525, 8.999364388223489, 9.364958555917556, 9.749435875862523, 10.12127089708159, 10.72119298751649, 11.28383669678605, 11.94409046108491, 12.26623200713094, 12.74375460755683, 13.31463678213126, 13.71340527979579
