| L(s) = 1 | + 0.228·2-s + 3-s − 1.94·4-s − 1.93·5-s + 0.228·6-s + 1.64·7-s − 0.901·8-s + 9-s − 0.443·10-s − 3.11·11-s − 1.94·12-s − 13-s + 0.375·14-s − 1.93·15-s + 3.68·16-s + 0.133·17-s + 0.228·18-s + 1.02·19-s + 3.77·20-s + 1.64·21-s − 0.711·22-s + 6.84·23-s − 0.901·24-s − 1.23·25-s − 0.228·26-s + 27-s − 3.20·28-s + ⋯ |
| L(s) = 1 | + 0.161·2-s + 0.577·3-s − 0.973·4-s − 0.867·5-s + 0.0932·6-s + 0.621·7-s − 0.318·8-s + 0.333·9-s − 0.140·10-s − 0.938·11-s − 0.562·12-s − 0.277·13-s + 0.100·14-s − 0.500·15-s + 0.922·16-s + 0.0323·17-s + 0.0538·18-s + 0.234·19-s + 0.844·20-s + 0.358·21-s − 0.151·22-s + 1.42·23-s − 0.184·24-s − 0.247·25-s − 0.0448·26-s + 0.192·27-s − 0.605·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 - 0.228T + 2T^{2} \) |
| 5 | \( 1 + 1.93T + 5T^{2} \) |
| 7 | \( 1 - 1.64T + 7T^{2} \) |
| 11 | \( 1 + 3.11T + 11T^{2} \) |
| 17 | \( 1 - 0.133T + 17T^{2} \) |
| 19 | \( 1 - 1.02T + 19T^{2} \) |
| 23 | \( 1 - 6.84T + 23T^{2} \) |
| 29 | \( 1 - 3.24T + 29T^{2} \) |
| 31 | \( 1 - 3.81T + 31T^{2} \) |
| 37 | \( 1 - 4.03T + 37T^{2} \) |
| 41 | \( 1 + 7.88T + 41T^{2} \) |
| 43 | \( 1 + 9.84T + 43T^{2} \) |
| 47 | \( 1 - 6.07T + 47T^{2} \) |
| 53 | \( 1 + 0.0869T + 53T^{2} \) |
| 59 | \( 1 + 3.11T + 59T^{2} \) |
| 61 | \( 1 + 9.30T + 61T^{2} \) |
| 67 | \( 1 + 5.15T + 67T^{2} \) |
| 71 | \( 1 - 2.32T + 71T^{2} \) |
| 73 | \( 1 + 11.9T + 73T^{2} \) |
| 79 | \( 1 + 7.62T + 79T^{2} \) |
| 83 | \( 1 - 6.31T + 83T^{2} \) |
| 89 | \( 1 + 5.62T + 89T^{2} \) |
| 97 | \( 1 + 8.05T + 97T^{2} \) |
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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
−8.100698617931285266966421885334, −7.65926389211957378473032067600, −6.79562273946857714263225146199, −5.58286217041047285459803923694, −4.81577450219311370824719469826, −4.42468885126181116373145856484, −3.38732978161375826540251111803, −2.77977541807516459686091593610, −1.32530274365573583681952001050, 0,
1.32530274365573583681952001050, 2.77977541807516459686091593610, 3.38732978161375826540251111803, 4.42468885126181116373145856484, 4.81577450219311370824719469826, 5.58286217041047285459803923694, 6.79562273946857714263225146199, 7.65926389211957378473032067600, 8.100698617931285266966421885334
