| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s + 9.14·5-s + 6·6-s − 7·7-s − 8·8-s + 9·9-s − 18.2·10-s + 15.2·11-s − 12·12-s + 4.29·13-s + 14·14-s − 27.4·15-s + 16·16-s − 45.3·17-s − 18·18-s + 19·19-s + 36.5·20-s + 21·21-s − 30.5·22-s − 155.·23-s + 24·24-s − 41.2·25-s − 8.59·26-s − 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.818·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.578·10-s + 0.419·11-s − 0.288·12-s + 0.0916·13-s + 0.267·14-s − 0.472·15-s + 0.250·16-s − 0.646·17-s − 0.235·18-s + 0.229·19-s + 0.409·20-s + 0.218·21-s − 0.296·22-s − 1.41·23-s + 0.204·24-s − 0.330·25-s − 0.0648·26-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 | \( 1 - 9.14T + 125T^{2} \) |
| 11 | \( 1 - 15.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 4.29T + 2.19e3T^{2} \) |
| 17 | \( 1 + 45.3T + 4.91e3T^{2} \) |
| 23 | \( 1 + 155.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 196.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 55.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 67.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 429.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 118.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 370.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 389.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 601.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 877.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 482.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 360.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.06e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 950.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 451.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 310.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.55e3T + 9.12e5T^{2} \) |
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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
−9.614313124924011019847746980078, −8.752046597058744714810555319711, −7.78680089985895265535755065310, −6.60489014416128675285336202800, −6.23151902622764996477252721893, −5.18075551362465971539090485533, −3.90271364470487325853467280366, −2.45932926458221738779920383413, −1.37659125665115121829996176608, 0,
1.37659125665115121829996176608, 2.45932926458221738779920383413, 3.90271364470487325853467280366, 5.18075551362465971539090485533, 6.23151902622764996477252721893, 6.60489014416128675285336202800, 7.78680089985895265535755065310, 8.752046597058744714810555319711, 9.614313124924011019847746980078
