| L(s) = 1 | + 2.22·3-s − 0.847·5-s − 7-s + 1.93·9-s + 4.11·11-s − 4.98·13-s − 1.88·15-s + 3.34·17-s + 19-s − 2.22·21-s − 8.84·23-s − 4.28·25-s − 2.36·27-s + 3.37·29-s + 7.98·31-s + 9.14·33-s + 0.847·35-s − 5.46·37-s − 11.0·39-s − 2.73·41-s − 8.71·43-s − 1.63·45-s − 8.05·47-s + 49-s + 7.42·51-s + 8.90·53-s − 3.48·55-s + ⋯ |
| L(s) = 1 | + 1.28·3-s − 0.378·5-s − 0.377·7-s + 0.644·9-s + 1.24·11-s − 1.38·13-s − 0.485·15-s + 0.811·17-s + 0.229·19-s − 0.484·21-s − 1.84·23-s − 0.856·25-s − 0.455·27-s + 0.626·29-s + 1.43·31-s + 1.59·33-s + 0.143·35-s − 0.897·37-s − 1.77·39-s − 0.427·41-s − 1.32·43-s − 0.244·45-s − 1.17·47-s + 0.142·49-s + 1.04·51-s + 1.22·53-s − 0.470·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 2.22T + 3T^{2} \) |
| 5 | \( 1 + 0.847T + 5T^{2} \) |
| 11 | \( 1 - 4.11T + 11T^{2} \) |
| 13 | \( 1 + 4.98T + 13T^{2} \) |
| 17 | \( 1 - 3.34T + 17T^{2} \) |
| 23 | \( 1 + 8.84T + 23T^{2} \) |
| 29 | \( 1 - 3.37T + 29T^{2} \) |
| 31 | \( 1 - 7.98T + 31T^{2} \) |
| 37 | \( 1 + 5.46T + 37T^{2} \) |
| 41 | \( 1 + 2.73T + 41T^{2} \) |
| 43 | \( 1 + 8.71T + 43T^{2} \) |
| 47 | \( 1 + 8.05T + 47T^{2} \) |
| 53 | \( 1 - 8.90T + 53T^{2} \) |
| 59 | \( 1 - 0.710T + 59T^{2} \) |
| 61 | \( 1 - 6.77T + 61T^{2} \) |
| 67 | \( 1 + 7.60T + 67T^{2} \) |
| 71 | \( 1 + 3.32T + 71T^{2} \) |
| 73 | \( 1 + 6.05T + 73T^{2} \) |
| 79 | \( 1 - 10.8T + 79T^{2} \) |
| 83 | \( 1 + 8.93T + 83T^{2} \) |
| 89 | \( 1 + 2.50T + 89T^{2} \) |
| 97 | \( 1 - 2.09T + 97T^{2} \) |
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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
−7.57932836795582002831158179992, −6.90524300904735665175886185585, −6.21847361491415927993855837537, −5.30573870854710181689010152707, −4.33936353170634305337860366620, −3.76267756254617296130416264432, −3.12244573732994146083431272757, −2.33217678400128101924486786854, −1.48954925994877350586784071088, 0,
1.48954925994877350586784071088, 2.33217678400128101924486786854, 3.12244573732994146083431272757, 3.76267756254617296130416264432, 4.33936353170634305337860366620, 5.30573870854710181689010152707, 6.21847361491415927993855837537, 6.90524300904735665175886185585, 7.57932836795582002831158179992
