| L(s) = 1 | + 10.5·2-s − 30.8·3-s + 80.0·4-s − 25·5-s − 326.·6-s − 29.6·7-s + 508.·8-s + 706.·9-s − 264.·10-s + 121.·11-s − 2.46e3·12-s + 795.·13-s − 314.·14-s + 770.·15-s + 2.82e3·16-s + 1.89e3·17-s + 7.47e3·18-s − 892.·19-s − 2.00e3·20-s + 914.·21-s + 1.28e3·22-s + 529·23-s − 1.56e4·24-s + 625·25-s + 8.41e3·26-s − 1.42e4·27-s − 2.37e3·28-s + ⋯ |
| L(s) = 1 | + 1.87·2-s − 1.97·3-s + 2.50·4-s − 0.447·5-s − 3.69·6-s − 0.229·7-s + 2.81·8-s + 2.90·9-s − 0.836·10-s + 0.302·11-s − 4.94·12-s + 1.30·13-s − 0.428·14-s + 0.884·15-s + 2.75·16-s + 1.58·17-s + 5.44·18-s − 0.566·19-s − 1.11·20-s + 0.452·21-s + 0.565·22-s + 0.208·23-s − 5.55·24-s + 0.200·25-s + 2.44·26-s − 3.77·27-s − 0.573·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.315924716\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.315924716\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 23 | \( 1 - 529T \) |
| good | 2 | \( 1 - 10.5T + 32T^{2} \) |
| 3 | \( 1 + 30.8T + 243T^{2} \) |
| 7 | \( 1 + 29.6T + 1.68e4T^{2} \) |
| 11 | \( 1 - 121.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 795.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.89e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 892.T + 2.47e6T^{2} \) |
| 29 | \( 1 - 428.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 369.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.09e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.27e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.20e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.58e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.63e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.10e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.64e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 8.91e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.58e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.05e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.53e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.11e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.63e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.36e4T + 8.58e9T^{2} \) |
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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
−12.58917015303631884008374668984, −11.74844511249505563000713382339, −11.17201933006302435652137088138, −10.24429103534365240324599997283, −7.45972423977695685982001119935, −6.32730710721513748875388304724, −5.78308965781526124347554799190, −4.63368121290443820077766871231, −3.65197476062493453990025237941, −1.18288539991737129115002278449,
1.18288539991737129115002278449, 3.65197476062493453990025237941, 4.63368121290443820077766871231, 5.78308965781526124347554799190, 6.32730710721513748875388304724, 7.45972423977695685982001119935, 10.24429103534365240324599997283, 11.17201933006302435652137088138, 11.74844511249505563000713382339, 12.58917015303631884008374668984
