| L(s) = 1 | + 2.47·2-s + 3-s + 4.11·4-s − 2.58·5-s + 2.47·6-s − 7-s + 5.22·8-s + 9-s − 6.39·10-s − 11-s + 4.11·12-s − 5.87·13-s − 2.47·14-s − 2.58·15-s + 4.70·16-s + 7.51·17-s + 2.47·18-s − 2.35·19-s − 10.6·20-s − 21-s − 2.47·22-s + 6.94·23-s + 5.22·24-s + 1.69·25-s − 14.5·26-s + 27-s − 4.11·28-s + ⋯ |
| L(s) = 1 | + 1.74·2-s + 0.577·3-s + 2.05·4-s − 1.15·5-s + 1.00·6-s − 0.377·7-s + 1.84·8-s + 0.333·9-s − 2.02·10-s − 0.301·11-s + 1.18·12-s − 1.62·13-s − 0.660·14-s − 0.668·15-s + 1.17·16-s + 1.82·17-s + 0.582·18-s − 0.540·19-s − 2.38·20-s − 0.218·21-s − 0.527·22-s + 1.44·23-s + 1.06·24-s + 0.339·25-s − 2.84·26-s + 0.192·27-s − 0.777·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.944315718\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.944315718\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 2.47T + 2T^{2} \) |
| 5 | \( 1 + 2.58T + 5T^{2} \) |
| 13 | \( 1 + 5.87T + 13T^{2} \) |
| 17 | \( 1 - 7.51T + 17T^{2} \) |
| 19 | \( 1 + 2.35T + 19T^{2} \) |
| 23 | \( 1 - 6.94T + 23T^{2} \) |
| 29 | \( 1 + 5.87T + 29T^{2} \) |
| 31 | \( 1 + 3.66T + 31T^{2} \) |
| 37 | \( 1 - 3.30T + 37T^{2} \) |
| 41 | \( 1 - 5.28T + 41T^{2} \) |
| 43 | \( 1 - 7.40T + 43T^{2} \) |
| 47 | \( 1 - 7.53T + 47T^{2} \) |
| 53 | \( 1 + 4.22T + 53T^{2} \) |
| 59 | \( 1 + 0.926T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 - 4.45T + 71T^{2} \) |
| 73 | \( 1 + 2.12T + 73T^{2} \) |
| 79 | \( 1 + 4.45T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−12.52404471879143102964196798798, −11.69988601541872017996300816489, −10.64145954225033776958527833627, −9.331081393855998552863132008413, −7.60328759004254524416029840779, −7.30779760444084841078467261484, −5.69653518869871741849245878104, −4.61930078937300391216756519466, −3.59901998740036571373751238840, −2.67958341900950104318063369812,
2.67958341900950104318063369812, 3.59901998740036571373751238840, 4.61930078937300391216756519466, 5.69653518869871741849245878104, 7.30779760444084841078467261484, 7.60328759004254524416029840779, 9.331081393855998552863132008413, 10.64145954225033776958527833627, 11.69988601541872017996300816489, 12.52404471879143102964196798798
