| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 5·5-s − 6·6-s + 3.49·7-s + 8·8-s + 9·9-s − 10·10-s − 36.4·11-s − 12·12-s + 21.3·13-s + 6.99·14-s + 15·15-s + 16·16-s − 11.0·17-s + 18·18-s + 141.·19-s − 20·20-s − 10.4·21-s − 72.8·22-s − 162.·23-s − 24·24-s + 25·25-s + 42.6·26-s − 27·27-s + 13.9·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s + 0.188·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.997·11-s − 0.288·12-s + 0.454·13-s + 0.133·14-s + 0.258·15-s + 0.250·16-s − 0.157·17-s + 0.235·18-s + 1.70·19-s − 0.223·20-s − 0.108·21-s − 0.705·22-s − 1.46·23-s − 0.204·24-s + 0.200·25-s + 0.321·26-s − 0.192·27-s + 0.0943·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3T \) |
| 5 | \( 1 + 5T \) |
| 31 | \( 1 + 31T \) |
| good | 7 | \( 1 - 3.49T + 343T^{2} \) |
| 11 | \( 1 + 36.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 21.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 11.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 141.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 162.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 51.4T + 2.43e4T^{2} \) |
| 37 | \( 1 - 111.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 373.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 200.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 302.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 179.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 219.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 701.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 282.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 369.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 354.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 917.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 459.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 280.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 490.T + 9.12e5T^{2} \) |
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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
−9.431162186787765067073036307138, −8.063271892820856500776789936690, −7.59162743527999674586900379873, −6.52165166366390985088631793618, −5.61133661033018231992677873001, −4.94508988625211948594681682747, −3.92533003341943290457952026303, −2.93224675664091625659337420493, −1.52077379362192804490032418928, 0,
1.52077379362192804490032418928, 2.93224675664091625659337420493, 3.92533003341943290457952026303, 4.94508988625211948594681682747, 5.61133661033018231992677873001, 6.52165166366390985088631793618, 7.59162743527999674586900379873, 8.063271892820856500776789936690, 9.431162186787765067073036307138
