| L(s) = 1 | − 2-s − 3.24·3-s + 4-s + 1.63·5-s + 3.24·6-s − 1.84·7-s − 8-s + 7.55·9-s − 1.63·10-s + 11-s − 3.24·12-s − 7.17·13-s + 1.84·14-s − 5.29·15-s + 16-s − 0.522·17-s − 7.55·18-s + 1.63·20-s + 5.99·21-s − 22-s + 4.05·23-s + 3.24·24-s − 2.34·25-s + 7.17·26-s − 14.7·27-s − 1.84·28-s − 7.91·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.87·3-s + 0.5·4-s + 0.729·5-s + 1.32·6-s − 0.697·7-s − 0.353·8-s + 2.51·9-s − 0.515·10-s + 0.301·11-s − 0.937·12-s − 1.98·13-s + 0.493·14-s − 1.36·15-s + 0.250·16-s − 0.126·17-s − 1.78·18-s + 0.364·20-s + 1.30·21-s − 0.213·22-s + 0.844·23-s + 0.663·24-s − 0.468·25-s + 1.40·26-s − 2.84·27-s − 0.348·28-s − 1.47·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 3.24T + 3T^{2} \) |
| 5 | \( 1 - 1.63T + 5T^{2} \) |
| 7 | \( 1 + 1.84T + 7T^{2} \) |
| 13 | \( 1 + 7.17T + 13T^{2} \) |
| 17 | \( 1 + 0.522T + 17T^{2} \) |
| 23 | \( 1 - 4.05T + 23T^{2} \) |
| 29 | \( 1 + 7.91T + 29T^{2} \) |
| 31 | \( 1 - 8.24T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 - 9.06T + 41T^{2} \) |
| 43 | \( 1 + 6.11T + 43T^{2} \) |
| 47 | \( 1 - 2.80T + 47T^{2} \) |
| 53 | \( 1 - 0.238T + 53T^{2} \) |
| 59 | \( 1 - 7.94T + 59T^{2} \) |
| 61 | \( 1 + 5.82T + 61T^{2} \) |
| 67 | \( 1 + 7.80T + 67T^{2} \) |
| 71 | \( 1 - 9.04T + 71T^{2} \) |
| 73 | \( 1 + 2.02T + 73T^{2} \) |
| 79 | \( 1 + 9.86T + 79T^{2} \) |
| 83 | \( 1 + 0.302T + 83T^{2} \) |
| 89 | \( 1 - 2.73T + 89T^{2} \) |
| 97 | \( 1 + 8.29T + 97T^{2} \) |
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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
−7.20823496754758375316720210336, −6.83134319230829874668025586339, −6.05363203817126666579677998402, −5.68190474298429242169574422833, −4.85899194007470877434710360948, −4.22879902084265739284284231010, −2.83826298128610302965901048619, −1.96181739918676075050799049435, −0.888782335200169761470929236369, 0,
0.888782335200169761470929236369, 1.96181739918676075050799049435, 2.83826298128610302965901048619, 4.22879902084265739284284231010, 4.85899194007470877434710360948, 5.68190474298429242169574422833, 6.05363203817126666579677998402, 6.83134319230829874668025586339, 7.20823496754758375316720210336
