| L(s) = 1 | − 1.13e4·2-s − 8.54e5·3-s + 9.41e7·4-s + 4.59e8·5-s + 9.65e9·6-s − 6.85e11·8-s − 1.17e11·9-s − 5.18e12·10-s − 7.40e12·11-s − 8.04e13·12-s + 3.03e13·13-s − 3.92e14·15-s + 4.58e15·16-s − 3.17e15·17-s + 1.32e15·18-s + 4.83e15·19-s + 4.32e16·20-s + 8.36e16·22-s + 1.44e17·23-s + 5.85e17·24-s − 8.72e16·25-s − 3.43e17·26-s + 8.24e17·27-s − 2.17e18·29-s + 4.43e18·30-s + 6.10e18·31-s − 2.88e19·32-s + ⋯ |
| L(s) = 1 | − 1.95·2-s − 0.928·3-s + 2.80·4-s + 0.840·5-s + 1.81·6-s − 3.52·8-s − 0.138·9-s − 1.64·10-s − 0.711·11-s − 2.60·12-s + 0.361·13-s − 0.780·15-s + 4.07·16-s − 1.32·17-s + 0.270·18-s + 0.501·19-s + 2.36·20-s + 1.38·22-s + 1.37·23-s + 3.27·24-s − 0.292·25-s − 0.705·26-s + 1.05·27-s − 1.14·29-s + 1.52·30-s + 1.39·31-s − 4.42·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + 1.13e4T + 3.35e7T^{2} \) |
| 3 | \( 1 + 8.54e5T + 8.47e11T^{2} \) |
| 5 | \( 1 - 4.59e8T + 2.98e17T^{2} \) |
| 11 | \( 1 + 7.40e12T + 1.08e26T^{2} \) |
| 13 | \( 1 - 3.03e13T + 7.05e27T^{2} \) |
| 17 | \( 1 + 3.17e15T + 5.77e30T^{2} \) |
| 19 | \( 1 - 4.83e15T + 9.30e31T^{2} \) |
| 23 | \( 1 - 1.44e17T + 1.10e34T^{2} \) |
| 29 | \( 1 + 2.17e18T + 3.63e36T^{2} \) |
| 31 | \( 1 - 6.10e18T + 1.92e37T^{2} \) |
| 37 | \( 1 + 1.64e19T + 1.60e39T^{2} \) |
| 41 | \( 1 + 1.90e20T + 2.08e40T^{2} \) |
| 43 | \( 1 - 7.54e19T + 6.86e40T^{2} \) |
| 47 | \( 1 - 3.46e20T + 6.34e41T^{2} \) |
| 53 | \( 1 - 7.87e20T + 1.27e43T^{2} \) |
| 59 | \( 1 - 1.22e20T + 1.86e44T^{2} \) |
| 61 | \( 1 - 2.62e22T + 4.29e44T^{2} \) |
| 67 | \( 1 + 5.94e22T + 4.48e45T^{2} \) |
| 71 | \( 1 + 1.33e23T + 1.91e46T^{2} \) |
| 73 | \( 1 - 8.09e21T + 3.82e46T^{2} \) |
| 79 | \( 1 - 2.65e23T + 2.75e47T^{2} \) |
| 83 | \( 1 - 1.12e24T + 9.48e47T^{2} \) |
| 89 | \( 1 + 1.23e24T + 5.42e48T^{2} \) |
| 97 | \( 1 - 7.14e24T + 4.66e49T^{2} \) |
| 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
−10.33907939049862373049070648023, −9.286808596825293026436482014976, −8.413292549603599060270392422422, −7.08820097765529613092547074021, −6.24375594026966501588081006073, −5.31723055312821960415926047533, −2.88406525148894716225189526883, −1.92632774286449641888577930629, −0.864617170542866848673996615928, 0,
0.864617170542866848673996615928, 1.92632774286449641888577930629, 2.88406525148894716225189526883, 5.31723055312821960415926047533, 6.24375594026966501588081006073, 7.08820097765529613092547074021, 8.413292549603599060270392422422, 9.286808596825293026436482014976, 10.33907939049862373049070648023
