| L(s) = 1 | + 0.643·2-s + 2.01·3-s − 1.58·4-s − 2.24·5-s + 1.29·6-s + 4.30·7-s − 2.30·8-s + 1.07·9-s − 1.44·10-s + 11-s − 3.20·12-s − 2.43·13-s + 2.77·14-s − 4.53·15-s + 1.68·16-s − 1.83·17-s + 0.691·18-s − 6.37·19-s + 3.55·20-s + 8.69·21-s + 0.643·22-s + 3.39·23-s − 4.65·24-s + 0.0383·25-s − 1.56·26-s − 3.88·27-s − 6.83·28-s + ⋯ |
| L(s) = 1 | + 0.454·2-s + 1.16·3-s − 0.792·4-s − 1.00·5-s + 0.530·6-s + 1.62·7-s − 0.815·8-s + 0.358·9-s − 0.456·10-s + 0.301·11-s − 0.924·12-s − 0.676·13-s + 0.740·14-s − 1.16·15-s + 0.421·16-s − 0.445·17-s + 0.162·18-s − 1.46·19-s + 0.796·20-s + 1.89·21-s + 0.137·22-s + 0.707·23-s − 0.950·24-s + 0.00767·25-s − 0.307·26-s − 0.748·27-s − 1.29·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9251 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9251 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 - 0.643T + 2T^{2} \) |
| 3 | \( 1 - 2.01T + 3T^{2} \) |
| 5 | \( 1 + 2.24T + 5T^{2} \) |
| 7 | \( 1 - 4.30T + 7T^{2} \) |
| 13 | \( 1 + 2.43T + 13T^{2} \) |
| 17 | \( 1 + 1.83T + 17T^{2} \) |
| 19 | \( 1 + 6.37T + 19T^{2} \) |
| 23 | \( 1 - 3.39T + 23T^{2} \) |
| 31 | \( 1 - 8.83T + 31T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 - 7.11T + 41T^{2} \) |
| 43 | \( 1 + 6.49T + 43T^{2} \) |
| 47 | \( 1 + 8.07T + 47T^{2} \) |
| 53 | \( 1 + 7.55T + 53T^{2} \) |
| 59 | \( 1 + 3.96T + 59T^{2} \) |
| 61 | \( 1 + 7.94T + 61T^{2} \) |
| 67 | \( 1 + 12.6T + 67T^{2} \) |
| 71 | \( 1 - 9.17T + 71T^{2} \) |
| 73 | \( 1 + 2.68T + 73T^{2} \) |
| 79 | \( 1 - 7.15T + 79T^{2} \) |
| 83 | \( 1 + 2.76T + 83T^{2} \) |
| 89 | \( 1 + 1.82T + 89T^{2} \) |
| 97 | \( 1 + 17.5T + 97T^{2} \) |
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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
−7.84112103219677247760634356817, −6.80298424287145182291982213699, −5.90839341206540513194529696432, −4.86604626095079192086717660719, −4.42160358173566464017063236845, −4.10958497527049304861201342723, −3.07407446589329387431461529461, −2.44267414391303925164260899307, −1.36747783225758614857861683440, 0,
1.36747783225758614857861683440, 2.44267414391303925164260899307, 3.07407446589329387431461529461, 4.10958497527049304861201342723, 4.42160358173566464017063236845, 4.86604626095079192086717660719, 5.90839341206540513194529696432, 6.80298424287145182291982213699, 7.84112103219677247760634356817
