| L(s) = 1 | + 3-s − 2·4-s + 5-s + 4.77·7-s + 9-s − 3.77·11-s − 2·12-s − 4.77·13-s + 15-s + 4·16-s + 6·17-s − 5.77·19-s − 2·20-s + 4.77·21-s + 25-s + 27-s − 9.54·28-s + 6·29-s + 5.77·31-s − 3.77·33-s + 4.77·35-s − 2·36-s − 1.22·37-s − 4.77·39-s − 9.77·41-s − 2.77·43-s + 7.54·44-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 0.447·5-s + 1.80·7-s + 0.333·9-s − 1.13·11-s − 0.577·12-s − 1.32·13-s + 0.258·15-s + 16-s + 1.45·17-s − 1.32·19-s − 0.447·20-s + 1.04·21-s + 0.200·25-s + 0.192·27-s − 1.80·28-s + 1.11·29-s + 1.03·31-s − 0.656·33-s + 0.806·35-s − 0.333·36-s − 0.201·37-s − 0.764·39-s − 1.52·41-s − 0.422·43-s + 1.13·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.463122155\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.463122155\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 7 | \( 1 - 4.77T + 7T^{2} \) |
| 11 | \( 1 + 3.77T + 11T^{2} \) |
| 13 | \( 1 + 4.77T + 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 5.77T + 19T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 5.77T + 31T^{2} \) |
| 37 | \( 1 + 1.22T + 37T^{2} \) |
| 41 | \( 1 + 9.77T + 41T^{2} \) |
| 43 | \( 1 + 2.77T + 43T^{2} \) |
| 47 | \( 1 - 7.54T + 47T^{2} \) |
| 53 | \( 1 + 13.5T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 8.54T + 61T^{2} \) |
| 67 | \( 1 - 3.22T + 67T^{2} \) |
| 71 | \( 1 - 9.77T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 13.7T + 79T^{2} \) |
| 83 | \( 1 - 13.5T + 83T^{2} \) |
| 89 | \( 1 + 7.54T + 89T^{2} \) |
| 97 | \( 1 - 17.5T + 97T^{2} \) |
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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
−7.980599019824832648967573264389, −7.56689496914881985922986367126, −6.45848292925154375035263371059, −5.35712784096746973939270971972, −4.93479619065096486123386870142, −4.61710405867366874501656357864, −3.53366585834806229294173903274, −2.54592596597594571718285443878, −1.87191625803479890455336666736, −0.77633078810472302419191624484,
0.77633078810472302419191624484, 1.87191625803479890455336666736, 2.54592596597594571718285443878, 3.53366585834806229294173903274, 4.61710405867366874501656357864, 4.93479619065096486123386870142, 5.35712784096746973939270971972, 6.45848292925154375035263371059, 7.56689496914881985922986367126, 7.980599019824832648967573264389
