| L(s) = 1 | − 2·4-s + 2·5-s − 7-s + 11-s + 4·16-s − 2·17-s + 7·19-s − 4·20-s − 7·23-s − 25-s + 2·28-s + 6·31-s − 2·35-s − 5·37-s + 3·41-s − 6·43-s − 2·44-s + 49-s − 3·53-s + 2·55-s − 7·61-s − 8·64-s + 8·67-s + 4·68-s − 71-s − 13·73-s − 14·76-s + ⋯ |
| L(s) = 1 | − 4-s + 0.894·5-s − 0.377·7-s + 0.301·11-s + 16-s − 0.485·17-s + 1.60·19-s − 0.894·20-s − 1.45·23-s − 1/5·25-s + 0.377·28-s + 1.07·31-s − 0.338·35-s − 0.821·37-s + 0.468·41-s − 0.914·43-s − 0.301·44-s + 1/7·49-s − 0.412·53-s + 0.269·55-s − 0.896·61-s − 64-s + 0.977·67-s + 0.485·68-s − 0.118·71-s − 1.52·73-s − 1.60·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−13.80819311403230, −13.52207928868793, −12.99690849008045, −12.43800338587852, −11.89244648685658, −11.63232146262564, −10.75924774312242, −10.20413241657955, −9.758929583301594, −9.648486672587118, −8.964098788208190, −8.579469154604241, −7.898634937417328, −7.517977310932052, −6.726242448096394, −6.206242789729527, −5.764017187020998, −5.277942653927860, −4.680504184646554, −4.164180156765229, −3.469005620255095, −3.053569021260410, −2.171043010796065, −1.570792835968616, −0.8352575576992271, 0,
0.8352575576992271, 1.570792835968616, 2.171043010796065, 3.053569021260410, 3.469005620255095, 4.164180156765229, 4.680504184646554, 5.277942653927860, 5.764017187020998, 6.206242789729527, 6.726242448096394, 7.517977310932052, 7.898634937417328, 8.579469154604241, 8.964098788208190, 9.648486672587118, 9.758929583301594, 10.20413241657955, 10.75924774312242, 11.63232146262564, 11.89244648685658, 12.43800338587852, 12.99690849008045, 13.52207928868793, 13.80819311403230
