| L(s) = 1 | − 4.48·2-s − 8.22·3-s + 12.0·4-s − 14.4·5-s + 36.8·6-s + 26.0·7-s − 18.2·8-s + 40.6·9-s + 64.9·10-s − 2.31·11-s − 99.3·12-s + 41.1·13-s − 116.·14-s + 119.·15-s − 14.8·16-s − 123.·17-s − 182.·18-s − 5.66·19-s − 174.·20-s − 214.·21-s + 10.3·22-s − 121.·23-s + 150.·24-s + 84.9·25-s − 184.·26-s − 112.·27-s + 314.·28-s + ⋯ |
| L(s) = 1 | − 1.58·2-s − 1.58·3-s + 1.50·4-s − 1.29·5-s + 2.50·6-s + 1.40·7-s − 0.806·8-s + 1.50·9-s + 2.05·10-s − 0.0633·11-s − 2.38·12-s + 0.878·13-s − 2.22·14-s + 2.05·15-s − 0.231·16-s − 1.75·17-s − 2.38·18-s − 0.0683·19-s − 1.95·20-s − 2.22·21-s + 0.100·22-s − 1.09·23-s + 1.27·24-s + 0.679·25-s − 1.39·26-s − 0.800·27-s + 2.12·28-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 | \( 1 + 4.48T + 8T^{2} \) |
| 3 | \( 1 + 8.22T + 27T^{2} \) |
| 5 | \( 1 + 14.4T + 125T^{2} \) |
| 7 | \( 1 - 26.0T + 343T^{2} \) |
| 11 | \( 1 + 2.31T + 1.33e3T^{2} \) |
| 13 | \( 1 - 41.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 123.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 5.66T + 6.85e3T^{2} \) |
| 23 | \( 1 + 121.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 9.00T + 2.43e4T^{2} \) |
| 37 | \( 1 - 249.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 17.8T + 6.89e4T^{2} \) |
| 43 | \( 1 + 59.1T + 7.95e4T^{2} \) |
| 47 | \( 1 - 282.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 459.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 179.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 161.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 123.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 769.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 289.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 402.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 678.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 626.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 796.T + 9.12e5T^{2} \) |
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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
−9.102067170998197175416545106228, −8.252173728639280949244076806736, −7.79064889451395221950518610519, −6.87062518655096482247048518468, −6.06408446694445112677949822563, −4.74598524233451924985618963047, −4.16435725623256421312493276515, −1.97256476319116325654892089442, −0.870985650916277584505208555997, 0,
0.870985650916277584505208555997, 1.97256476319116325654892089442, 4.16435725623256421312493276515, 4.74598524233451924985618963047, 6.06408446694445112677949822563, 6.87062518655096482247048518468, 7.79064889451395221950518610519, 8.252173728639280949244076806736, 9.102067170998197175416545106228
