| L(s) = 1 | + 2.05·3-s − 2.64·5-s + 3.00·7-s + 1.21·9-s − 5.46·11-s + 3.55·13-s − 5.43·15-s + 0.722·17-s + 6.30·19-s + 6.16·21-s − 7.59·23-s + 2.01·25-s − 3.67·27-s + 0.328·29-s − 6.90·31-s − 11.2·33-s − 7.95·35-s + 2.38·37-s + 7.29·39-s + 0.0570·41-s − 3.29·43-s − 3.20·45-s + 11.6·47-s + 2.02·49-s + 1.48·51-s − 9.18·53-s + 14.4·55-s + ⋯ |
| L(s) = 1 | + 1.18·3-s − 1.18·5-s + 1.13·7-s + 0.403·9-s − 1.64·11-s + 0.985·13-s − 1.40·15-s + 0.175·17-s + 1.44·19-s + 1.34·21-s − 1.58·23-s + 0.403·25-s − 0.706·27-s + 0.0609·29-s − 1.23·31-s − 1.95·33-s − 1.34·35-s + 0.392·37-s + 1.16·39-s + 0.00890·41-s − 0.502·43-s − 0.478·45-s + 1.70·47-s + 0.289·49-s + 0.207·51-s − 1.26·53-s + 1.95·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 - 2.05T + 3T^{2} \) |
| 5 | \( 1 + 2.64T + 5T^{2} \) |
| 7 | \( 1 - 3.00T + 7T^{2} \) |
| 11 | \( 1 + 5.46T + 11T^{2} \) |
| 13 | \( 1 - 3.55T + 13T^{2} \) |
| 17 | \( 1 - 0.722T + 17T^{2} \) |
| 19 | \( 1 - 6.30T + 19T^{2} \) |
| 23 | \( 1 + 7.59T + 23T^{2} \) |
| 29 | \( 1 - 0.328T + 29T^{2} \) |
| 31 | \( 1 + 6.90T + 31T^{2} \) |
| 37 | \( 1 - 2.38T + 37T^{2} \) |
| 41 | \( 1 - 0.0570T + 41T^{2} \) |
| 43 | \( 1 + 3.29T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 + 9.18T + 53T^{2} \) |
| 59 | \( 1 + 0.912T + 59T^{2} \) |
| 61 | \( 1 + 9.05T + 61T^{2} \) |
| 67 | \( 1 - 5.43T + 67T^{2} \) |
| 71 | \( 1 + 9.61T + 71T^{2} \) |
| 73 | \( 1 - 1.13T + 73T^{2} \) |
| 79 | \( 1 + 7.13T + 79T^{2} \) |
| 83 | \( 1 - 3.59T + 83T^{2} \) |
| 89 | \( 1 + 3.86T + 89T^{2} \) |
| 97 | \( 1 - 7.98T + 97T^{2} \) |
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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
−7.58979094442581614029041760458, −7.45356273213136517923097367743, −5.94056698796164993272095261903, −5.34217552691871240907847240792, −4.48876511240691475093045557099, −3.75185369525987599749858285846, −3.18978113401691534601656054065, −2.34735099386616720523773519997, −1.43708664527620617793455464066, 0,
1.43708664527620617793455464066, 2.34735099386616720523773519997, 3.18978113401691534601656054065, 3.75185369525987599749858285846, 4.48876511240691475093045557099, 5.34217552691871240907847240792, 5.94056698796164993272095261903, 7.45356273213136517923097367743, 7.58979094442581614029041760458
