| L(s) = 1 | + 4.62·2-s + 17.6·3-s − 10.5·4-s − 25·5-s + 81.5·6-s + 46.5·7-s − 197.·8-s + 67.2·9-s − 115.·10-s + 121·11-s − 186.·12-s − 639.·13-s + 215.·14-s − 440.·15-s − 574.·16-s + 2.16e3·17-s + 311.·18-s + 361·19-s + 264.·20-s + 820.·21-s + 560.·22-s + 197.·23-s − 3.47e3·24-s + 625·25-s − 2.96e3·26-s − 3.09e3·27-s − 492.·28-s + ⋯ |
| L(s) = 1 | + 0.818·2-s + 1.12·3-s − 0.330·4-s − 0.447·5-s + 0.924·6-s + 0.359·7-s − 1.08·8-s + 0.276·9-s − 0.365·10-s + 0.301·11-s − 0.373·12-s − 1.04·13-s + 0.293·14-s − 0.505·15-s − 0.560·16-s + 1.81·17-s + 0.226·18-s + 0.229·19-s + 0.147·20-s + 0.405·21-s + 0.246·22-s + 0.0779·23-s − 1.23·24-s + 0.200·25-s − 0.858·26-s − 0.817·27-s − 0.118·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 - 4.62T + 32T^{2} \) |
| 3 | \( 1 - 17.6T + 243T^{2} \) |
| 7 | \( 1 - 46.5T + 1.68e4T^{2} \) |
| 13 | \( 1 + 639.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.16e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 197.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.85e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.97e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.70e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.04e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.76e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.91e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.09e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 6.38e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.41e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.50e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.44e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.76e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.50e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.38e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.21e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.07e5T + 8.58e9T^{2} \) |
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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
−8.594823018469322520901804986279, −8.083204941288060074400543750760, −7.27250144604318299364005780928, −6.04794229789829803373536609374, −5.03501401129471387894865448143, −4.37567374106515474724094162717, −3.20507132823762805892820492472, −2.95870009405153940820412856281, −1.42274023537907194976872377962, 0,
1.42274023537907194976872377962, 2.95870009405153940820412856281, 3.20507132823762805892820492472, 4.37567374106515474724094162717, 5.03501401129471387894865448143, 6.04794229789829803373536609374, 7.27250144604318299364005780928, 8.083204941288060074400543750760, 8.594823018469322520901804986279
