| L(s) = 1 | − 2.49·2-s + 4.20·4-s − 7-s − 5.49·8-s − 0.713·11-s − 0.777·13-s + 2.49·14-s + 5.26·16-s − 4.06·17-s + 2.35·19-s + 1.77·22-s − 3.20·23-s + 1.93·26-s − 4.20·28-s + 3.20·29-s + 5.75·31-s − 2.14·32-s + 10.1·34-s + 2.63·37-s − 5.85·38-s − 8.54·41-s + 7.71·43-s − 2.99·44-s + 7.98·46-s + 3.75·47-s + 49-s − 3.26·52-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 2.10·4-s − 0.377·7-s − 1.94·8-s − 0.215·11-s − 0.215·13-s + 0.665·14-s + 1.31·16-s − 0.985·17-s + 0.539·19-s + 0.378·22-s − 0.668·23-s + 0.379·26-s − 0.794·28-s + 0.595·29-s + 1.03·31-s − 0.378·32-s + 1.73·34-s + 0.432·37-s − 0.949·38-s − 1.33·41-s + 1.17·43-s − 0.452·44-s + 1.17·46-s + 0.548·47-s + 0.142·49-s − 0.453·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 2 | \( 1 + 2.49T + 2T^{2} \) |
| 11 | \( 1 + 0.713T + 11T^{2} \) |
| 13 | \( 1 + 0.777T + 13T^{2} \) |
| 17 | \( 1 + 4.06T + 17T^{2} \) |
| 19 | \( 1 - 2.35T + 19T^{2} \) |
| 23 | \( 1 + 3.20T + 23T^{2} \) |
| 29 | \( 1 - 3.20T + 29T^{2} \) |
| 31 | \( 1 - 5.75T + 31T^{2} \) |
| 37 | \( 1 - 2.63T + 37T^{2} \) |
| 41 | \( 1 + 8.54T + 41T^{2} \) |
| 43 | \( 1 - 7.71T + 43T^{2} \) |
| 47 | \( 1 - 3.75T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 - 1.34T + 61T^{2} \) |
| 67 | \( 1 - 9.94T + 67T^{2} \) |
| 71 | \( 1 + 0.567T + 71T^{2} \) |
| 73 | \( 1 + 6.83T + 73T^{2} \) |
| 79 | \( 1 - 6.83T + 79T^{2} \) |
| 83 | \( 1 - 6.24T + 83T^{2} \) |
| 89 | \( 1 + 2.12T + 89T^{2} \) |
| 97 | \( 1 + 4.85T + 97T^{2} \) |
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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
−8.177663465150945917928096921685, −7.38505722648940634223584367728, −6.71141569573416076029643458610, −6.18747723527267889762432671139, −5.11306937717436361672572677257, −4.05851549059155064022807055363, −2.85821174318149857188141486139, −2.20987536136794925464248301401, −1.08404170461200630417862878712, 0,
1.08404170461200630417862878712, 2.20987536136794925464248301401, 2.85821174318149857188141486139, 4.05851549059155064022807055363, 5.11306937717436361672572677257, 6.18747723527267889762432671139, 6.71141569573416076029643458610, 7.38505722648940634223584367728, 8.177663465150945917928096921685
