| L(s) = 1 | + 42.0·2-s + 1.25e3·4-s − 625·5-s + 2.40e3·7-s + 3.13e4·8-s − 2.62e4·10-s + 746.·11-s − 1.39e5·13-s + 1.01e5·14-s + 6.76e5·16-s − 6.57e5·17-s + 5.86e4·19-s − 7.86e5·20-s + 3.13e4·22-s − 2.05e6·23-s + 3.90e5·25-s − 5.85e6·26-s + 3.02e6·28-s + 2.78e6·29-s − 4.13e6·31-s + 1.23e7·32-s − 2.76e7·34-s − 1.50e6·35-s − 3.47e6·37-s + 2.46e6·38-s − 1.96e7·40-s − 6.46e6·41-s + ⋯ |
| L(s) = 1 | + 1.85·2-s + 2.45·4-s − 0.447·5-s + 0.377·7-s + 2.70·8-s − 0.831·10-s + 0.0153·11-s − 1.35·13-s + 0.702·14-s + 2.58·16-s − 1.91·17-s + 0.103·19-s − 1.09·20-s + 0.0285·22-s − 1.53·23-s + 0.200·25-s − 2.51·26-s + 0.928·28-s + 0.730·29-s − 0.804·31-s + 2.08·32-s − 3.55·34-s − 0.169·35-s − 0.304·37-s + 0.192·38-s − 1.21·40-s − 0.357·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 625T \) |
| 7 | \( 1 - 2.40e3T \) |
| good | 2 | \( 1 - 42.0T + 512T^{2} \) |
| 11 | \( 1 - 746.T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.39e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 6.57e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 5.86e4T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.05e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.78e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 4.13e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 3.47e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 6.46e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.69e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 5.51e6T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.00e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 3.57e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.95e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.00e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.54e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 4.48e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.37e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 6.74e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 5.23e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.00e9T + 7.60e17T^{2} \) |
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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
−9.967077926009845161391535666422, −8.401279308230185668311816365483, −7.27394267319601216437375937907, −6.59248644576821164699407024647, −5.41881228336484319106344835026, −4.57829622528875754637731116254, −3.92304727808932936453962383673, −2.62973776203302289034250603237, −1.90139302121829854285751001864, 0,
1.90139302121829854285751001864, 2.62973776203302289034250603237, 3.92304727808932936453962383673, 4.57829622528875754637731116254, 5.41881228336484319106344835026, 6.59248644576821164699407024647, 7.27394267319601216437375937907, 8.401279308230185668311816365483, 9.967077926009845161391535666422
