| L(s) = 1 | − 3-s − 1.13·5-s − 1.30·7-s + 9-s − 2.48·11-s + 3.75·13-s + 1.13·15-s − 0.436·17-s − 0.587·19-s + 1.30·21-s − 3.21·23-s − 3.70·25-s − 27-s + 4.25·29-s + 7.37·31-s + 2.48·33-s + 1.47·35-s − 6.01·37-s − 3.75·39-s + 10.9·41-s + 0.537·43-s − 1.13·45-s − 4.83·47-s − 5.30·49-s + 0.436·51-s + 7.95·53-s + 2.82·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.508·5-s − 0.492·7-s + 0.333·9-s − 0.749·11-s + 1.04·13-s + 0.293·15-s − 0.105·17-s − 0.134·19-s + 0.284·21-s − 0.671·23-s − 0.741·25-s − 0.192·27-s + 0.789·29-s + 1.32·31-s + 0.432·33-s + 0.250·35-s − 0.989·37-s − 0.601·39-s + 1.71·41-s + 0.0819·43-s − 0.169·45-s − 0.705·47-s − 0.757·49-s + 0.0611·51-s + 1.09·53-s + 0.381·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 173 | \( 1 - T \) |
| good | 5 | \( 1 + 1.13T + 5T^{2} \) |
| 7 | \( 1 + 1.30T + 7T^{2} \) |
| 11 | \( 1 + 2.48T + 11T^{2} \) |
| 13 | \( 1 - 3.75T + 13T^{2} \) |
| 17 | \( 1 + 0.436T + 17T^{2} \) |
| 19 | \( 1 + 0.587T + 19T^{2} \) |
| 23 | \( 1 + 3.21T + 23T^{2} \) |
| 29 | \( 1 - 4.25T + 29T^{2} \) |
| 31 | \( 1 - 7.37T + 31T^{2} \) |
| 37 | \( 1 + 6.01T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 - 0.537T + 43T^{2} \) |
| 47 | \( 1 + 4.83T + 47T^{2} \) |
| 53 | \( 1 - 7.95T + 53T^{2} \) |
| 59 | \( 1 - 4.53T + 59T^{2} \) |
| 61 | \( 1 + 1.40T + 61T^{2} \) |
| 67 | \( 1 + 8.68T + 67T^{2} \) |
| 71 | \( 1 + 8.38T + 71T^{2} \) |
| 73 | \( 1 - 15.0T + 73T^{2} \) |
| 79 | \( 1 + 9.66T + 79T^{2} \) |
| 83 | \( 1 - 9.17T + 83T^{2} \) |
| 89 | \( 1 + 1.50T + 89T^{2} \) |
| 97 | \( 1 - 8.68T + 97T^{2} \) |
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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
−7.52860652101150929431878952811, −6.59032547578326818260823499599, −6.16415472969542669521937170190, −5.46364317328774565952587305869, −4.59681267205695420072658153828, −3.95441556258685375879380726590, −3.18066534869245285326584787981, −2.24029517774325040735478657763, −1.03767794694925073725380025373, 0,
1.03767794694925073725380025373, 2.24029517774325040735478657763, 3.18066534869245285326584787981, 3.95441556258685375879380726590, 4.59681267205695420072658153828, 5.46364317328774565952587305869, 6.16415472969542669521937170190, 6.59032547578326818260823499599, 7.52860652101150929431878952811
