| L(s) = 1 | − 0.291·2-s + 3-s − 1.91·4-s − 3.42·5-s − 0.291·6-s − 3.90·7-s + 1.14·8-s + 9-s + 0.998·10-s + 4.06·11-s − 1.91·12-s − 13-s + 1.14·14-s − 3.42·15-s + 3.49·16-s − 0.751·17-s − 0.291·18-s + 5.37·19-s + 6.54·20-s − 3.90·21-s − 1.18·22-s − 8.24·23-s + 1.14·24-s + 6.69·25-s + 0.291·26-s + 27-s + 7.48·28-s + ⋯ |
| L(s) = 1 | − 0.206·2-s + 0.577·3-s − 0.957·4-s − 1.52·5-s − 0.119·6-s − 1.47·7-s + 0.404·8-s + 0.333·9-s + 0.315·10-s + 1.22·11-s − 0.552·12-s − 0.277·13-s + 0.305·14-s − 0.883·15-s + 0.873·16-s − 0.182·17-s − 0.0688·18-s + 1.23·19-s + 1.46·20-s − 0.852·21-s − 0.252·22-s − 1.72·23-s + 0.233·24-s + 1.33·25-s + 0.0572·26-s + 0.192·27-s + 1.41·28-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 | \( 1 + 0.291T + 2T^{2} \) |
| 5 | \( 1 + 3.42T + 5T^{2} \) |
| 7 | \( 1 + 3.90T + 7T^{2} \) |
| 11 | \( 1 - 4.06T + 11T^{2} \) |
| 17 | \( 1 + 0.751T + 17T^{2} \) |
| 19 | \( 1 - 5.37T + 19T^{2} \) |
| 23 | \( 1 + 8.24T + 23T^{2} \) |
| 29 | \( 1 - 8.16T + 29T^{2} \) |
| 31 | \( 1 - 1.89T + 31T^{2} \) |
| 37 | \( 1 + 4.14T + 37T^{2} \) |
| 41 | \( 1 - 0.461T + 41T^{2} \) |
| 43 | \( 1 - 0.847T + 43T^{2} \) |
| 47 | \( 1 - 8.10T + 47T^{2} \) |
| 53 | \( 1 - 2.55T + 53T^{2} \) |
| 59 | \( 1 + 5.39T + 59T^{2} \) |
| 61 | \( 1 + 1.43T + 61T^{2} \) |
| 67 | \( 1 - 5.33T + 67T^{2} \) |
| 71 | \( 1 + 1.28T + 71T^{2} \) |
| 73 | \( 1 - 5.97T + 73T^{2} \) |
| 79 | \( 1 + 8.83T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 + 4.26T + 97T^{2} \) |
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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
−8.204419794813243268293206835876, −7.47039717912523636198349772271, −6.82645641070677175950750007989, −5.97560918560154784227516762769, −4.76640926396665946314944997755, −3.91130010184282084580483853591, −3.70836141278086766340959105160, −2.80410606138575924118932411040, −1.05463247670934054990169606954, 0,
1.05463247670934054990169606954, 2.80410606138575924118932411040, 3.70836141278086766340959105160, 3.91130010184282084580483853591, 4.76640926396665946314944997755, 5.97560918560154784227516762769, 6.82645641070677175950750007989, 7.47039717912523636198349772271, 8.204419794813243268293206835876
