| L(s) = 1 | − 4·2-s − 2.48·3-s + 16·4-s + 25·5-s + 9.93·6-s − 252.·7-s − 64·8-s − 236.·9-s − 100·10-s − 739.·11-s − 39.7·12-s + 876.·13-s + 1.00e3·14-s − 62.0·15-s + 256·16-s − 1.36e3·17-s + 947.·18-s + 1.72e3·19-s + 400·20-s + 626.·21-s + 2.95e3·22-s − 529·23-s + 158.·24-s + 625·25-s − 3.50e3·26-s + 1.19e3·27-s − 4.03e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.159·3-s + 0.5·4-s + 0.447·5-s + 0.112·6-s − 1.94·7-s − 0.353·8-s − 0.974·9-s − 0.316·10-s − 1.84·11-s − 0.0796·12-s + 1.43·13-s + 1.37·14-s − 0.0712·15-s + 0.250·16-s − 1.14·17-s + 0.689·18-s + 1.09·19-s + 0.223·20-s + 0.310·21-s + 1.30·22-s − 0.208·23-s + 0.0563·24-s + 0.200·25-s − 1.01·26-s + 0.314·27-s − 0.972·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.5438721557\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5438721557\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 5 | \( 1 - 25T \) |
| 23 | \( 1 + 529T \) |
| good | 3 | \( 1 + 2.48T + 243T^{2} \) |
| 7 | \( 1 + 252.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 739.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 876.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.36e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.72e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 3.06e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.42e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.04e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.63e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 8.73e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.89e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.89e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.97e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.90e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 9.39e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.89e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.53e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.76e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.76e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.32e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.13e4T + 8.58e9T^{2} \) |
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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
−10.93680364483236959528965937825, −10.42131160077200183235240595696, −9.321969633833439353213223283940, −8.648927356531074494961812345030, −7.32222095929009796636911334311, −6.19759414824420650973667070984, −5.56512780361802919978146842680, −3.37279809146272384538592055529, −2.50050709954954397295413728214, −0.45951503934233084012503678500,
0.45951503934233084012503678500, 2.50050709954954397295413728214, 3.37279809146272384538592055529, 5.56512780361802919978146842680, 6.19759414824420650973667070984, 7.32222095929009796636911334311, 8.648927356531074494961812345030, 9.321969633833439353213223283940, 10.42131160077200183235240595696, 10.93680364483236959528965937825
