| L(s) = 1 | − 133.·2-s + 64.3·3-s + 9.58e3·4-s + 1.95e4·5-s − 8.57e3·6-s + 1.77e5·7-s − 1.85e5·8-s − 1.59e6·9-s − 2.60e6·10-s − 3.36e5·11-s + 6.16e5·12-s + 1.41e7·13-s − 2.36e7·14-s + 1.25e6·15-s − 5.37e7·16-s + 1.10e8·17-s + 2.12e8·18-s − 1.18e8·19-s + 1.87e8·20-s + 1.14e7·21-s + 4.47e7·22-s − 7.03e8·23-s − 1.19e7·24-s − 8.37e8·25-s − 1.89e9·26-s − 2.04e8·27-s + 1.69e9·28-s + ⋯ |
| L(s) = 1 | − 1.47·2-s + 0.0509·3-s + 1.16·4-s + 0.559·5-s − 0.0750·6-s + 0.569·7-s − 0.250·8-s − 0.997·9-s − 0.824·10-s − 0.0571·11-s + 0.0596·12-s + 0.815·13-s − 0.839·14-s + 0.0285·15-s − 0.801·16-s + 1.11·17-s + 1.46·18-s − 0.575·19-s + 0.654·20-s + 0.0290·21-s + 0.0842·22-s − 0.990·23-s − 0.0127·24-s − 0.686·25-s − 1.20·26-s − 0.101·27-s + 0.666·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.9840262625\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9840262625\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 197 | \( 1 - 5.84e13T \) |
| good | 2 | \( 1 + 133.T + 8.19e3T^{2} \) |
| 3 | \( 1 - 64.3T + 1.59e6T^{2} \) |
| 5 | \( 1 - 1.95e4T + 1.22e9T^{2} \) |
| 7 | \( 1 - 1.77e5T + 9.68e10T^{2} \) |
| 11 | \( 1 + 3.36e5T + 3.45e13T^{2} \) |
| 13 | \( 1 - 1.41e7T + 3.02e14T^{2} \) |
| 17 | \( 1 - 1.10e8T + 9.90e15T^{2} \) |
| 19 | \( 1 + 1.18e8T + 4.20e16T^{2} \) |
| 23 | \( 1 + 7.03e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 3.44e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 5.07e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 2.95e10T + 2.43e20T^{2} \) |
| 41 | \( 1 - 3.95e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 5.83e10T + 1.71e21T^{2} \) |
| 47 | \( 1 - 8.53e10T + 5.46e21T^{2} \) |
| 53 | \( 1 + 2.30e11T + 2.60e22T^{2} \) |
| 59 | \( 1 - 2.95e11T + 1.04e23T^{2} \) |
| 61 | \( 1 - 3.76e11T + 1.61e23T^{2} \) |
| 67 | \( 1 - 5.34e11T + 5.48e23T^{2} \) |
| 71 | \( 1 - 1.73e12T + 1.16e24T^{2} \) |
| 73 | \( 1 - 6.32e11T + 1.67e24T^{2} \) |
| 79 | \( 1 + 1.64e12T + 4.66e24T^{2} \) |
| 83 | \( 1 + 7.58e11T + 8.87e24T^{2} \) |
| 89 | \( 1 - 1.01e11T + 2.19e25T^{2} \) |
| 97 | \( 1 + 1.06e12T + 6.73e25T^{2} \) |
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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
−9.917110897775969962423120305841, −9.228999269754609107886690237397, −8.212401890454571390395868178109, −7.75244428763011171391723583953, −6.28122600901542719184391858681, −5.45246981919792835667509007961, −3.85512020974786597001394527392, −2.36883831769383456422097539829, −1.57310229908007807017715488488, −0.52716100692339385437479282583,
0.52716100692339385437479282583, 1.57310229908007807017715488488, 2.36883831769383456422097539829, 3.85512020974786597001394527392, 5.45246981919792835667509007961, 6.28122600901542719184391858681, 7.75244428763011171391723583953, 8.212401890454571390395868178109, 9.228999269754609107886690237397, 9.917110897775969962423120305841
