| L(s) = 1 | + 2·2-s + 2·4-s − 2·11-s + 13-s − 4·16-s − 2·17-s − 7·19-s − 4·22-s + 6·23-s + 2·26-s − 4·29-s − 3·31-s − 8·32-s − 4·34-s − 2·37-s − 14·38-s − 8·41-s + 11·43-s − 4·44-s + 12·46-s + 4·47-s + 2·52-s + 2·53-s − 8·58-s − 10·59-s + 5·61-s − 6·62-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 0.603·11-s + 0.277·13-s − 16-s − 0.485·17-s − 1.60·19-s − 0.852·22-s + 1.25·23-s + 0.392·26-s − 0.742·29-s − 0.538·31-s − 1.41·32-s − 0.685·34-s − 0.328·37-s − 2.27·38-s − 1.24·41-s + 1.67·43-s − 0.603·44-s + 1.76·46-s + 0.583·47-s + 0.277·52-s + 0.274·53-s − 1.05·58-s − 1.30·59-s + 0.640·61-s − 0.762·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 297675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.269474739\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.269474739\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 15 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.78817586119718, −12.39781377734905, −12.05009640919881, −11.19024035932769, −11.02935904063287, −10.74651987978054, −10.06485046769490, −9.419971560984912, −8.878501784177785, −8.682490343064728, −7.988436135842234, −7.378177278355402, −6.915942003089849, −6.459722072829738, −6.022497023020833, −5.445533884869863, −5.081559282627139, −4.590112065129518, −4.023190830245376, −3.679803255933695, −3.065243833985124, −2.394922024401613, −2.164963847406803, −1.280720264193294, −0.2979745006703295,
0.2979745006703295, 1.280720264193294, 2.164963847406803, 2.394922024401613, 3.065243833985124, 3.679803255933695, 4.023190830245376, 4.590112065129518, 5.081559282627139, 5.445533884869863, 6.022497023020833, 6.459722072829738, 6.915942003089849, 7.378177278355402, 7.988436135842234, 8.682490343064728, 8.878501784177785, 9.419971560984912, 10.06485046769490, 10.74651987978054, 11.02935904063287, 11.19024035932769, 12.05009640919881, 12.39781377734905, 12.78817586119718
