| L(s) = 1 | − 0.618·2-s − 2·3-s − 1.61·4-s + 1.23·6-s − 3·7-s + 2.23·8-s + 9-s + 3.23·12-s − 2.38·13-s + 1.85·14-s + 1.85·16-s + 1.47·17-s − 0.618·18-s − 6.23·19-s + 6·21-s − 3.38·23-s − 4.47·24-s + 1.47·26-s + 4·27-s + 4.85·28-s − 1.52·29-s + 4.23·31-s − 5.61·32-s − 0.909·34-s − 1.61·36-s − 10.7·37-s + 3.85·38-s + ⋯ |
| L(s) = 1 | − 0.437·2-s − 1.15·3-s − 0.809·4-s + 0.504·6-s − 1.13·7-s + 0.790·8-s + 0.333·9-s + 0.934·12-s − 0.660·13-s + 0.495·14-s + 0.463·16-s + 0.357·17-s − 0.145·18-s − 1.43·19-s + 1.30·21-s − 0.705·23-s − 0.912·24-s + 0.288·26-s + 0.769·27-s + 0.917·28-s − 0.283·29-s + 0.760·31-s − 0.993·32-s − 0.156·34-s − 0.269·36-s − 1.76·37-s + 0.625·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 53 | \( 1 + T \) |
| good | 2 | \( 1 + 0.618T + 2T^{2} \) |
| 3 | \( 1 + 2T + 3T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 2.38T + 13T^{2} \) |
| 17 | \( 1 - 1.47T + 17T^{2} \) |
| 19 | \( 1 + 6.23T + 19T^{2} \) |
| 23 | \( 1 + 3.38T + 23T^{2} \) |
| 29 | \( 1 + 1.52T + 29T^{2} \) |
| 31 | \( 1 - 4.23T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 + 1.09T + 41T^{2} \) |
| 43 | \( 1 - 0.236T + 43T^{2} \) |
| 47 | \( 1 + 11.2T + 47T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 - 6.14T + 61T^{2} \) |
| 67 | \( 1 + 0.145T + 67T^{2} \) |
| 71 | \( 1 + 4.38T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 + 9.94T + 83T^{2} \) |
| 89 | \( 1 - 0.618T + 89T^{2} \) |
| 97 | \( 1 + 15.5T + 97T^{2} \) |
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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
−7.08447348646786401979462397378, −6.62257180226094632127499934572, −5.83817665932347132915016098611, −5.25631191530780437388139807876, −4.48685028225218570473488338691, −3.77034727732211643526188082049, −2.75907549635824004950488188758, −1.46745035877378628273348350851, 0, 0,
1.46745035877378628273348350851, 2.75907549635824004950488188758, 3.77034727732211643526188082049, 4.48685028225218570473488338691, 5.25631191530780437388139807876, 5.83817665932347132915016098611, 6.62257180226094632127499934572, 7.08447348646786401979462397378
