| L(s) = 1 | − 353.·2-s − 4.03e3·3-s + 9.22e4·4-s + 2.46e5·5-s + 1.42e6·6-s + 5.07e5·7-s − 2.10e7·8-s + 1.93e6·9-s − 8.71e7·10-s − 1.32e7·11-s − 3.72e8·12-s + 8.34e7·13-s − 1.79e8·14-s − 9.95e8·15-s + 4.41e9·16-s + 1.62e9·17-s − 6.84e8·18-s − 6.97e9·19-s + 2.27e10·20-s − 2.04e9·21-s + 4.67e9·22-s − 1.57e10·23-s + 8.49e10·24-s + 3.02e10·25-s − 2.94e10·26-s + 5.00e10·27-s + 4.68e10·28-s + ⋯ |
| L(s) = 1 | − 1.95·2-s − 1.06·3-s + 2.81·4-s + 1.41·5-s + 2.08·6-s + 0.232·7-s − 3.54·8-s + 0.134·9-s − 2.75·10-s − 0.204·11-s − 3.00·12-s + 0.368·13-s − 0.454·14-s − 1.50·15-s + 4.11·16-s + 0.962·17-s − 0.263·18-s − 1.79·19-s + 3.97·20-s − 0.248·21-s + 0.399·22-s − 0.966·23-s + 3.77·24-s + 0.992·25-s − 0.720·26-s + 0.921·27-s + 0.655·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 - 1.72e10T \) |
| good | 2 | \( 1 + 353.T + 3.27e4T^{2} \) |
| 3 | \( 1 + 4.03e3T + 1.43e7T^{2} \) |
| 5 | \( 1 - 2.46e5T + 3.05e10T^{2} \) |
| 7 | \( 1 - 5.07e5T + 4.74e12T^{2} \) |
| 11 | \( 1 + 1.32e7T + 4.17e15T^{2} \) |
| 13 | \( 1 - 8.34e7T + 5.11e16T^{2} \) |
| 17 | \( 1 - 1.62e9T + 2.86e18T^{2} \) |
| 19 | \( 1 + 6.97e9T + 1.51e19T^{2} \) |
| 23 | \( 1 + 1.57e10T + 2.66e20T^{2} \) |
| 31 | \( 1 + 1.47e11T + 2.34e22T^{2} \) |
| 37 | \( 1 - 7.69e11T + 3.33e23T^{2} \) |
| 41 | \( 1 - 2.02e12T + 1.55e24T^{2} \) |
| 43 | \( 1 - 4.99e11T + 3.17e24T^{2} \) |
| 47 | \( 1 + 5.62e12T + 1.20e25T^{2} \) |
| 53 | \( 1 - 1.08e13T + 7.31e25T^{2} \) |
| 59 | \( 1 + 1.11e13T + 3.65e26T^{2} \) |
| 61 | \( 1 + 2.72e13T + 6.02e26T^{2} \) |
| 67 | \( 1 + 5.10e12T + 2.46e27T^{2} \) |
| 71 | \( 1 - 1.13e14T + 5.87e27T^{2} \) |
| 73 | \( 1 + 4.22e13T + 8.90e27T^{2} \) |
| 79 | \( 1 - 2.88e13T + 2.91e28T^{2} \) |
| 83 | \( 1 - 2.94e14T + 6.11e28T^{2} \) |
| 89 | \( 1 + 4.13e14T + 1.74e29T^{2} \) |
| 97 | \( 1 + 1.35e15T + 6.33e29T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.54457995739477483167570971245, −11.16395603531822367039430494146, −10.41037061066461840047820280639, −9.416020397150507723033975717824, −8.094734529425091446642481063832, −6.41161095766834790983614518520, −5.78054172446436851065043716778, −2.36099735394131031127178286284, −1.27368556337408571519567074772, 0,
1.27368556337408571519567074772, 2.36099735394131031127178286284, 5.78054172446436851065043716778, 6.41161095766834790983614518520, 8.094734529425091446642481063832, 9.416020397150507723033975717824, 10.41037061066461840047820280639, 11.16395603531822367039430494146, 12.54457995739477483167570971245
