| L(s) = 1 | + 2.49·5-s − 0.917·7-s − 3.69·11-s − 5.81·13-s + 0.867·17-s + 6.52·19-s + 4·23-s + 1.23·25-s − 7.72·29-s + 2.14·31-s − 2.29·35-s + 2.47·37-s + 9.58·41-s − 9.58·43-s − 1.65·47-s − 6.15·49-s + 3.39·53-s − 9.22·55-s + 12.7·59-s − 0.0231·61-s − 14.5·65-s + 5.32·67-s + 11.8·71-s − 15.2·73-s + 3.39·77-s − 8.40·79-s − 1.96·83-s + ⋯ |
| L(s) = 1 | + 1.11·5-s − 0.346·7-s − 1.11·11-s − 1.61·13-s + 0.210·17-s + 1.49·19-s + 0.834·23-s + 0.246·25-s − 1.43·29-s + 0.385·31-s − 0.387·35-s + 0.406·37-s + 1.49·41-s − 1.46·43-s − 0.241·47-s − 0.879·49-s + 0.466·53-s − 1.24·55-s + 1.65·59-s − 0.00296·61-s − 1.79·65-s + 0.650·67-s + 1.40·71-s − 1.78·73-s + 0.386·77-s − 0.946·79-s − 0.215·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2.49T + 5T^{2} \) |
| 7 | \( 1 + 0.917T + 7T^{2} \) |
| 11 | \( 1 + 3.69T + 11T^{2} \) |
| 13 | \( 1 + 5.81T + 13T^{2} \) |
| 17 | \( 1 - 0.867T + 17T^{2} \) |
| 19 | \( 1 - 6.52T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 7.72T + 29T^{2} \) |
| 31 | \( 1 - 2.14T + 31T^{2} \) |
| 37 | \( 1 - 2.47T + 37T^{2} \) |
| 41 | \( 1 - 9.58T + 41T^{2} \) |
| 43 | \( 1 + 9.58T + 43T^{2} \) |
| 47 | \( 1 + 1.65T + 47T^{2} \) |
| 53 | \( 1 - 3.39T + 53T^{2} \) |
| 59 | \( 1 - 12.7T + 59T^{2} \) |
| 61 | \( 1 + 0.0231T + 61T^{2} \) |
| 67 | \( 1 - 5.32T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 + 8.40T + 79T^{2} \) |
| 83 | \( 1 + 1.96T + 83T^{2} \) |
| 89 | \( 1 + 2.79T + 89T^{2} \) |
| 97 | \( 1 + 2.26T + 97T^{2} \) |
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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
−7.44166004728983828714758945117, −6.74747506024820907388419690498, −5.85951527965114274953070835503, −5.22281794701898443932555358518, −4.99639436980184845848476970269, −3.73658463048441612020983547773, −2.75132115437229098395546682756, −2.41178357261953016740689264437, −1.29922943881697012518481396061, 0,
1.29922943881697012518481396061, 2.41178357261953016740689264437, 2.75132115437229098395546682756, 3.73658463048441612020983547773, 4.99639436980184845848476970269, 5.22281794701898443932555358518, 5.85951527965114274953070835503, 6.74747506024820907388419690498, 7.44166004728983828714758945117
