| L(s) = 1 | − 2-s − 3-s + 4-s − 2.73·5-s + 6-s + 3.23·7-s − 8-s + 9-s + 2.73·10-s + 4.82·11-s − 12-s + 2.97·13-s − 3.23·14-s + 2.73·15-s + 16-s − 4.98·17-s − 18-s − 19-s − 2.73·20-s − 3.23·21-s − 4.82·22-s − 4.19·23-s + 24-s + 2.50·25-s − 2.97·26-s − 27-s + 3.23·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.22·5-s + 0.408·6-s + 1.22·7-s − 0.353·8-s + 0.333·9-s + 0.866·10-s + 1.45·11-s − 0.288·12-s + 0.826·13-s − 0.864·14-s + 0.707·15-s + 0.250·16-s − 1.20·17-s − 0.235·18-s − 0.229·19-s − 0.612·20-s − 0.705·21-s − 1.02·22-s − 0.875·23-s + 0.204·24-s + 0.500·25-s − 0.584·26-s − 0.192·27-s + 0.611·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| good | 5 | \( 1 + 2.73T + 5T^{2} \) |
| 7 | \( 1 - 3.23T + 7T^{2} \) |
| 11 | \( 1 - 4.82T + 11T^{2} \) |
| 13 | \( 1 - 2.97T + 13T^{2} \) |
| 17 | \( 1 + 4.98T + 17T^{2} \) |
| 23 | \( 1 + 4.19T + 23T^{2} \) |
| 29 | \( 1 - 3.90T + 29T^{2} \) |
| 31 | \( 1 - 0.453T + 31T^{2} \) |
| 37 | \( 1 + 4.36T + 37T^{2} \) |
| 41 | \( 1 + 9.85T + 41T^{2} \) |
| 43 | \( 1 - 9.43T + 43T^{2} \) |
| 47 | \( 1 - 1.30T + 47T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 + 13.2T + 61T^{2} \) |
| 67 | \( 1 - 7.76T + 67T^{2} \) |
| 71 | \( 1 - 3.85T + 71T^{2} \) |
| 73 | \( 1 + 8.13T + 73T^{2} \) |
| 79 | \( 1 + 6.64T + 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 + 0.669T + 89T^{2} \) |
| 97 | \( 1 + 6.36T + 97T^{2} \) |
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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
−7.80988223854044495781888227320, −7.07688189393112816452219567992, −6.48312220990045205931808796659, −5.75882341612645587748963840173, −4.49181032833646758304293610641, −4.29263461427309189247557741779, −3.34578640726972263667293714921, −1.90225576460213147792789596452, −1.21106111255796264483080429688, 0,
1.21106111255796264483080429688, 1.90225576460213147792789596452, 3.34578640726972263667293714921, 4.29263461427309189247557741779, 4.49181032833646758304293610641, 5.75882341612645587748963840173, 6.48312220990045205931808796659, 7.07688189393112816452219567992, 7.80988223854044495781888227320
