| L(s) = 1 | + 3.44·2-s + 3·3-s + 3.89·4-s − 5·5-s + 10.3·6-s − 14.1·8-s + 9·9-s − 17.2·10-s − 5.84·11-s + 11.6·12-s − 3.12·13-s − 15·15-s − 79.9·16-s − 62.1·17-s + 31.0·18-s + 17.6·19-s − 19.4·20-s − 20.1·22-s − 112.·23-s − 42.4·24-s + 25·25-s − 10.7·26-s + 27·27-s − 164.·29-s − 51.7·30-s + 20.2·31-s − 162.·32-s + ⋯ |
| L(s) = 1 | + 1.21·2-s + 0.577·3-s + 0.486·4-s − 0.447·5-s + 0.703·6-s − 0.625·8-s + 0.333·9-s − 0.545·10-s − 0.160·11-s + 0.280·12-s − 0.0667·13-s − 0.258·15-s − 1.24·16-s − 0.886·17-s + 0.406·18-s + 0.212·19-s − 0.217·20-s − 0.195·22-s − 1.01·23-s − 0.361·24-s + 0.200·25-s − 0.0814·26-s + 0.192·27-s − 1.05·29-s − 0.314·30-s + 0.117·31-s − 0.898·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 3T \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 3.44T + 8T^{2} \) |
| 11 | \( 1 + 5.84T + 1.33e3T^{2} \) |
| 13 | \( 1 + 3.12T + 2.19e3T^{2} \) |
| 17 | \( 1 + 62.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 17.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 112.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 164.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 20.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 300.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 83.8T + 6.89e4T^{2} \) |
| 43 | \( 1 + 44.5T + 7.95e4T^{2} \) |
| 47 | \( 1 - 88.6T + 1.03e5T^{2} \) |
| 53 | \( 1 + 363.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 660.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 805.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 510.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 615.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 30.6T + 3.89e5T^{2} \) |
| 79 | \( 1 - 235.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 229.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.46e3T + 7.04e5T^{2} \) |
| 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
−9.433452754860636125538637852185, −8.706367676524001665999933296999, −7.74457059182801060959936831783, −6.79133305509100288127586120452, −5.81035291381217053551635350786, −4.78731193275618642709179724220, −3.99015148140149828481422327872, −3.17506575025664656211138791098, −2.03418902156462835054619168296, 0,
2.03418902156462835054619168296, 3.17506575025664656211138791098, 3.99015148140149828481422327872, 4.78731193275618642709179724220, 5.81035291381217053551635350786, 6.79133305509100288127586120452, 7.74457059182801060959936831783, 8.706367676524001665999933296999, 9.433452754860636125538637852185
