| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 11.4·5-s − 6·6-s − 7·7-s + 8·8-s + 9·9-s − 22.9·10-s + 58.8·11-s − 12·12-s + 5.39·13-s − 14·14-s + 34.4·15-s + 16·16-s − 105.·17-s + 18·18-s − 19·19-s − 45.9·20-s + 21·21-s + 117.·22-s + 206.·23-s − 24·24-s + 6.70·25-s + 10.7·26-s − 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.02·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.725·10-s + 1.61·11-s − 0.288·12-s + 0.115·13-s − 0.267·14-s + 0.592·15-s + 0.250·16-s − 1.51·17-s + 0.235·18-s − 0.229·19-s − 0.513·20-s + 0.218·21-s + 1.14·22-s + 1.87·23-s − 0.204·24-s + 0.0536·25-s + 0.0814·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 + 7T \) |
| 19 | \( 1 + 19T \) |
| good | 5 | \( 1 + 11.4T + 125T^{2} \) |
| 11 | \( 1 - 58.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 5.39T + 2.19e3T^{2} \) |
| 17 | \( 1 + 105.T + 4.91e3T^{2} \) |
| 23 | \( 1 - 206.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 201.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 156.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 325.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 145.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 507.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 311.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 408.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 350.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 500.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 527.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 342.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 286.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.13e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 672.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 407.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 844.T + 9.12e5T^{2} \) |
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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
−9.356436898872963060835960361960, −8.694955104976785283579712258750, −7.33416666108680395492177942562, −6.77464350387983514999091537537, −6.00433645222493515803387875879, −4.69444779252259583024100655696, −4.09658707766537415518688149846, −3.15066277590454660756381357952, −1.48711479671144251441890311651, 0,
1.48711479671144251441890311651, 3.15066277590454660756381357952, 4.09658707766537415518688149846, 4.69444779252259583024100655696, 6.00433645222493515803387875879, 6.77464350387983514999091537537, 7.33416666108680395492177942562, 8.694955104976785283579712258750, 9.356436898872963060835960361960
