| L(s) = 1 | − 2.32·2-s − 3-s + 3.42·4-s − 5-s + 2.32·6-s − 3.29·7-s − 3.32·8-s + 9-s + 2.32·10-s + 0.936·11-s − 3.42·12-s + 5.10·13-s + 7.66·14-s + 15-s + 0.882·16-s − 2.12·17-s − 2.32·18-s − 6.67·19-s − 3.42·20-s + 3.29·21-s − 2.18·22-s + 1.78·23-s + 3.32·24-s + 25-s − 11.8·26-s − 27-s − 11.2·28-s + ⋯ |
| L(s) = 1 | − 1.64·2-s − 0.577·3-s + 1.71·4-s − 0.447·5-s + 0.950·6-s − 1.24·7-s − 1.17·8-s + 0.333·9-s + 0.736·10-s + 0.282·11-s − 0.988·12-s + 1.41·13-s + 2.04·14-s + 0.258·15-s + 0.220·16-s − 0.515·17-s − 0.549·18-s − 1.53·19-s − 0.765·20-s + 0.718·21-s − 0.464·22-s + 0.371·23-s + 0.677·24-s + 0.200·25-s − 2.33·26-s − 0.192·27-s − 2.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2476192895\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2476192895\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 + 2.32T + 2T^{2} \) |
| 7 | \( 1 + 3.29T + 7T^{2} \) |
| 11 | \( 1 - 0.936T + 11T^{2} \) |
| 13 | \( 1 - 5.10T + 13T^{2} \) |
| 17 | \( 1 + 2.12T + 17T^{2} \) |
| 19 | \( 1 + 6.67T + 19T^{2} \) |
| 23 | \( 1 - 1.78T + 23T^{2} \) |
| 29 | \( 1 + 8.02T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 - 7.21T + 37T^{2} \) |
| 41 | \( 1 - 4.13T + 41T^{2} \) |
| 43 | \( 1 + 7.24T + 43T^{2} \) |
| 47 | \( 1 - 5.38T + 47T^{2} \) |
| 53 | \( 1 - 14.1T + 53T^{2} \) |
| 59 | \( 1 + 9.57T + 59T^{2} \) |
| 61 | \( 1 - 8.62T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 - 4.10T + 71T^{2} \) |
| 73 | \( 1 - 14.2T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 + 8.83T + 83T^{2} \) |
| 89 | \( 1 - 6.25T + 89T^{2} \) |
| 97 | \( 1 + 15.2T + 97T^{2} \) |
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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
−8.222365022161096086897195615529, −7.38854300057829372240214837210, −6.79466938988649011658309010676, −6.28119814970901593646579940913, −5.61294323845046295638837963630, −4.17426013683542024396947239293, −3.66324598931782776628340229562, −2.44665876995168047189373092309, −1.45850621912651771917683104160, −0.35855307550006409900927256338,
0.35855307550006409900927256338, 1.45850621912651771917683104160, 2.44665876995168047189373092309, 3.66324598931782776628340229562, 4.17426013683542024396947239293, 5.61294323845046295638837963630, 6.28119814970901593646579940913, 6.79466938988649011658309010676, 7.38854300057829372240214837210, 8.222365022161096086897195615529
