| L(s) = 1 | − 39.2·2-s + 1.02e3·4-s + 625·5-s − 2.40e3·7-s − 2.01e4·8-s − 2.45e4·10-s − 2.80e4·11-s − 1.38e5·13-s + 9.41e4·14-s + 2.63e5·16-s + 9.16e4·17-s + 4.84e4·19-s + 6.40e5·20-s + 1.10e6·22-s + 1.37e6·23-s + 3.90e5·25-s + 5.43e6·26-s − 2.46e6·28-s + 3.84e6·29-s − 7.06e6·31-s − 4.65e4·32-s − 3.59e6·34-s − 1.50e6·35-s + 8.92e6·37-s − 1.89e6·38-s − 1.25e7·40-s + 1.03e7·41-s + ⋯ |
| L(s) = 1 | − 1.73·2-s + 2.00·4-s + 0.447·5-s − 0.377·7-s − 1.73·8-s − 0.774·10-s − 0.578·11-s − 1.34·13-s + 0.654·14-s + 1.00·16-s + 0.266·17-s + 0.0852·19-s + 0.895·20-s + 1.00·22-s + 1.02·23-s + 0.200·25-s + 2.33·26-s − 0.756·28-s + 1.00·29-s − 1.37·31-s − 0.00784·32-s − 0.460·34-s − 0.169·35-s + 0.782·37-s − 0.147·38-s − 0.776·40-s + 0.573·41-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 | \( 1 + 39.2T + 512T^{2} \) |
| 11 | \( 1 + 2.80e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.38e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 9.16e4T + 1.18e11T^{2} \) |
| 19 | \( 1 - 4.84e4T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.37e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 3.84e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 7.06e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 8.92e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.03e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.48e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.79e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.86e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 4.07e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 2.10e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 4.12e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.45e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.20e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.88e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 6.49e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 6.31e8T + 3.50e17T^{2} \) |
| 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
−9.529926991753673401926145825282, −8.958798665454191775886489739799, −7.75181074461078914109951729945, −7.19455851909712716526031228806, −6.10960683413165327726538418903, −4.87290866606344729493325032800, −2.96229214630390096444131852386, −2.16340944345734252717458307741, −0.959118277880721762417210956129, 0,
0.959118277880721762417210956129, 2.16340944345734252717458307741, 2.96229214630390096444131852386, 4.87290866606344729493325032800, 6.10960683413165327726538418903, 7.19455851909712716526031228806, 7.75181074461078914109951729945, 8.958798665454191775886489739799, 9.529926991753673401926145825282
