| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s + 6·11-s − 4·13-s + 16-s + 17-s − 19-s − 20-s − 6·22-s + 3·23-s − 4·25-s + 4·26-s + 6·29-s − 32-s − 34-s − 3·37-s + 38-s + 40-s − 12·41-s + 3·43-s + 6·44-s − 3·46-s − 10·47-s + 4·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s + 1.80·11-s − 1.10·13-s + 1/4·16-s + 0.242·17-s − 0.229·19-s − 0.223·20-s − 1.27·22-s + 0.625·23-s − 4/5·25-s + 0.784·26-s + 1.11·29-s − 0.176·32-s − 0.171·34-s − 0.493·37-s + 0.162·38-s + 0.158·40-s − 1.87·41-s + 0.457·43-s + 0.904·44-s − 0.442·46-s − 1.45·47-s + 0.565·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14994 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14994 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.295662918\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.295662918\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 13 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−16.19877880642306, −15.54032957919395, −14.84704913481524, −14.65690676036828, −13.94445287884393, −13.28812536117671, −12.29675256069346, −12.13983315354434, −11.52295547712857, −11.10078069290145, −10.08934763838443, −9.856564932971706, −9.214983893677079, −8.484192519353614, −8.184852633729435, −7.185846388420738, −6.879114873904592, −6.327064721718560, −5.371431290327158, −4.674964144291861, −3.859784797531763, −3.286626208673547, −2.286537525771261, −1.495871820311182, −0.5766580168615331,
0.5766580168615331, 1.495871820311182, 2.286537525771261, 3.286626208673547, 3.859784797531763, 4.674964144291861, 5.371431290327158, 6.327064721718560, 6.879114873904592, 7.185846388420738, 8.184852633729435, 8.484192519353614, 9.214983893677079, 9.856564932971706, 10.08934763838443, 11.10078069290145, 11.52295547712857, 12.13983315354434, 12.29675256069346, 13.28812536117671, 13.94445287884393, 14.65690676036828, 14.84704913481524, 15.54032957919395, 16.19877880642306
