| L(s) = 1 | − 5.13·2-s + 4.07·3-s + 18.3·4-s + 9.39·5-s − 20.9·6-s + 18.3·7-s − 52.9·8-s − 10.3·9-s − 48.2·10-s − 11·11-s + 74.7·12-s − 94.2·14-s + 38.3·15-s + 125.·16-s − 89.6·17-s + 53.2·18-s − 27.3·19-s + 172.·20-s + 74.9·21-s + 56.4·22-s − 16.8·23-s − 216.·24-s − 36.6·25-s − 152.·27-s + 336.·28-s + 92.4·29-s − 196.·30-s + ⋯ |
| L(s) = 1 | − 1.81·2-s + 0.784·3-s + 2.29·4-s + 0.840·5-s − 1.42·6-s + 0.992·7-s − 2.34·8-s − 0.384·9-s − 1.52·10-s − 0.301·11-s + 1.79·12-s − 1.80·14-s + 0.659·15-s + 1.95·16-s − 1.27·17-s + 0.696·18-s − 0.329·19-s + 1.92·20-s + 0.778·21-s + 0.546·22-s − 0.152·23-s − 1.83·24-s − 0.293·25-s − 1.08·27-s + 2.27·28-s + 0.592·29-s − 1.19·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 5.13T + 8T^{2} \) |
| 3 | \( 1 - 4.07T + 27T^{2} \) |
| 5 | \( 1 - 9.39T + 125T^{2} \) |
| 7 | \( 1 - 18.3T + 343T^{2} \) |
| 17 | \( 1 + 89.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 27.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 16.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 92.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 156.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 302.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 402.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 4.90T + 7.95e4T^{2} \) |
| 47 | \( 1 - 23.9T + 1.03e5T^{2} \) |
| 53 | \( 1 - 455.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 656.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 785.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 238.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 102.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 709.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 629.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.06e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 918.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 865.T + 9.12e5T^{2} \) |
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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
−8.639536857679027098047769221220, −7.968401860476150325687048157672, −7.36408523486157918603227977425, −6.34769741720126245184535905507, −5.58392247448247001814907375543, −4.24330143343319196310101446944, −2.62662842484344942430460809762, −2.23712163381336083532233699509, −1.33414107609518902066213719577, 0,
1.33414107609518902066213719577, 2.23712163381336083532233699509, 2.62662842484344942430460809762, 4.24330143343319196310101446944, 5.58392247448247001814907375543, 6.34769741720126245184535905507, 7.36408523486157918603227977425, 7.968401860476150325687048157672, 8.639536857679027098047769221220
