| L(s) = 1 | + 0.923·2-s + 8.16·3-s − 7.14·4-s + 4.12·5-s + 7.54·6-s + 13.2·7-s − 13.9·8-s + 39.7·9-s + 3.81·10-s − 49.1·11-s − 58.3·12-s − 59.3·13-s + 12.2·14-s + 33.7·15-s + 44.2·16-s − 27.3·17-s + 36.6·18-s − 112.·19-s − 29.5·20-s + 108.·21-s − 45.3·22-s + 114.·23-s − 114.·24-s − 107.·25-s − 54.8·26-s + 103.·27-s − 94.9·28-s + ⋯ |
| L(s) = 1 | + 0.326·2-s + 1.57·3-s − 0.893·4-s + 0.369·5-s + 0.513·6-s + 0.717·7-s − 0.618·8-s + 1.47·9-s + 0.120·10-s − 1.34·11-s − 1.40·12-s − 1.26·13-s + 0.234·14-s + 0.580·15-s + 0.691·16-s − 0.389·17-s + 0.480·18-s − 1.36·19-s − 0.329·20-s + 1.12·21-s − 0.439·22-s + 1.03·23-s − 0.971·24-s − 0.863·25-s − 0.413·26-s + 0.739·27-s − 0.640·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 31 | \( 1 \) |
| good | 2 | \( 1 - 0.923T + 8T^{2} \) |
| 3 | \( 1 - 8.16T + 27T^{2} \) |
| 5 | \( 1 - 4.12T + 125T^{2} \) |
| 7 | \( 1 - 13.2T + 343T^{2} \) |
| 11 | \( 1 + 49.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 59.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 27.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 112.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 114.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 23.8T + 2.43e4T^{2} \) |
| 37 | \( 1 + 280.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 469.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 215.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 381.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 131.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 844.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 375.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 174.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 48.7T + 3.57e5T^{2} \) |
| 73 | \( 1 + 515.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 53.3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 792.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 5.42T + 7.04e5T^{2} \) |
| 97 | \( 1 + 386.T + 9.12e5T^{2} \) |
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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
−9.026597134659926110430963198812, −8.435769299888706199221158150879, −7.87802356327703792823761199075, −6.89555285532405139425384996376, −5.32001457072717515043230096169, −4.78750166881315354260003852781, −3.74687682886969271624280948022, −2.67784402820759871388445968586, −1.94753780635918798760890864596, 0,
1.94753780635918798760890864596, 2.67784402820759871388445968586, 3.74687682886969271624280948022, 4.78750166881315354260003852781, 5.32001457072717515043230096169, 6.89555285532405139425384996376, 7.87802356327703792823761199075, 8.435769299888706199221158150879, 9.026597134659926110430963198812
