| L(s) = 1 | + 2.59·2-s + 3-s + 4.74·4-s + 2.59·6-s + 3.28·7-s + 7.12·8-s + 9-s + 4.30·11-s + 4.74·12-s − 3.46·13-s + 8.52·14-s + 9.02·16-s − 5.44·17-s + 2.59·18-s − 7.63·19-s + 3.28·21-s + 11.1·22-s − 5.04·23-s + 7.12·24-s − 8.98·26-s + 27-s + 15.5·28-s − 3.12·29-s − 2.06·31-s + 9.18·32-s + 4.30·33-s − 14.1·34-s + ⋯ |
| L(s) = 1 | + 1.83·2-s + 0.577·3-s + 2.37·4-s + 1.06·6-s + 1.24·7-s + 2.52·8-s + 0.333·9-s + 1.29·11-s + 1.36·12-s − 0.959·13-s + 2.27·14-s + 2.25·16-s − 1.32·17-s + 0.612·18-s − 1.75·19-s + 0.716·21-s + 2.38·22-s − 1.05·23-s + 1.45·24-s − 1.76·26-s + 0.192·27-s + 2.94·28-s − 0.579·29-s − 0.371·31-s + 1.62·32-s + 0.749·33-s − 2.42·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.215393937\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.215393937\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 2 | \( 1 - 2.59T + 2T^{2} \) |
| 7 | \( 1 - 3.28T + 7T^{2} \) |
| 11 | \( 1 - 4.30T + 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 + 5.44T + 17T^{2} \) |
| 19 | \( 1 + 7.63T + 19T^{2} \) |
| 23 | \( 1 + 5.04T + 23T^{2} \) |
| 29 | \( 1 + 3.12T + 29T^{2} \) |
| 31 | \( 1 + 2.06T + 31T^{2} \) |
| 37 | \( 1 + 1.89T + 37T^{2} \) |
| 41 | \( 1 - 3.89T + 41T^{2} \) |
| 43 | \( 1 + 3.20T + 43T^{2} \) |
| 47 | \( 1 - 6.28T + 47T^{2} \) |
| 53 | \( 1 + 2.51T + 53T^{2} \) |
| 59 | \( 1 - 7.72T + 59T^{2} \) |
| 61 | \( 1 + 2.95T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 + 4.72T + 71T^{2} \) |
| 73 | \( 1 + 1.64T + 73T^{2} \) |
| 79 | \( 1 - 13.7T + 79T^{2} \) |
| 83 | \( 1 + 5.01T + 83T^{2} \) |
| 89 | \( 1 + 9.00T + 89T^{2} \) |
| 97 | \( 1 - 2.57T + 97T^{2} \) |
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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
−9.079799361310410998750429545617, −8.311553151871954525169256294252, −7.36096026466434618268319823137, −6.68248730896595240549359437194, −5.92607795350072501210051348497, −4.80741584127087054261904246173, −4.32608016554883566239321854279, −3.70766970038151483459633409674, −2.23752470807957915845228057743, −1.93324782642117804631262741868,
1.93324782642117804631262741868, 2.23752470807957915845228057743, 3.70766970038151483459633409674, 4.32608016554883566239321854279, 4.80741584127087054261904246173, 5.92607795350072501210051348497, 6.68248730896595240549359437194, 7.36096026466434618268319823137, 8.311553151871954525169256294252, 9.079799361310410998750429545617
