| L(s) = 1 | − 4·2-s − 10.5·3-s + 16·4-s + 25·5-s + 42.1·6-s + 156.·7-s − 64·8-s − 132.·9-s − 100·10-s − 397.·11-s − 168.·12-s − 71.8·13-s − 625.·14-s − 263.·15-s + 256·16-s + 221.·17-s + 528.·18-s − 173.·19-s + 400·20-s − 1.64e3·21-s + 1.58e3·22-s + 529·23-s + 673.·24-s + 625·25-s + 287.·26-s + 3.94e3·27-s + 2.50e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.675·3-s + 0.5·4-s + 0.447·5-s + 0.477·6-s + 1.20·7-s − 0.353·8-s − 0.543·9-s − 0.316·10-s − 0.990·11-s − 0.337·12-s − 0.117·13-s − 0.852·14-s − 0.302·15-s + 0.250·16-s + 0.185·17-s + 0.384·18-s − 0.110·19-s + 0.223·20-s − 0.814·21-s + 0.700·22-s + 0.208·23-s + 0.238·24-s + 0.200·25-s + 0.0833·26-s + 1.04·27-s + 0.603·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + 4T \) |
| 5 | \( 1 - 25T \) |
| 23 | \( 1 - 529T \) |
| good | 3 | \( 1 + 10.5T + 243T^{2} \) |
| 7 | \( 1 - 156.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 397.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 71.8T + 3.71e5T^{2} \) |
| 17 | \( 1 - 221.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 173.T + 2.47e6T^{2} \) |
| 29 | \( 1 - 6.96e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.00e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.23e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 420.T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.42e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.70e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 6.10e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.32e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.93e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.63e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 9.48e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.76e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.64e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.08e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.96e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.40e5T + 8.58e9T^{2} \) |
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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
−10.77561341693149960735590236781, −10.13269444966921758802733330619, −8.724332034260981078606541786075, −8.057407065832032467977675786475, −6.85029743204636348980689541743, −5.61785304449320236799386416915, −4.87206561129133824150849184274, −2.77443967185949700124127870306, −1.44744975642420044948402398842, 0,
1.44744975642420044948402398842, 2.77443967185949700124127870306, 4.87206561129133824150849184274, 5.61785304449320236799386416915, 6.85029743204636348980689541743, 8.057407065832032467977675786475, 8.724332034260981078606541786075, 10.13269444966921758802733330619, 10.77561341693149960735590236781
