| L(s) = 1 | + 128·2-s − 219.·3-s + 1.63e4·4-s − 2.49e5·5-s − 2.81e4·6-s + 2.09e6·8-s − 1.43e7·9-s − 3.19e7·10-s + 3.30e7·11-s − 3.59e6·12-s − 5.07e7·13-s + 5.48e7·15-s + 2.68e8·16-s + 3.03e9·17-s − 1.83e9·18-s + 5.51e9·19-s − 4.08e9·20-s + 4.23e9·22-s + 3.19e9·23-s − 4.60e8·24-s + 3.17e10·25-s − 6.49e9·26-s + 6.29e9·27-s − 1.57e11·29-s + 7.01e9·30-s + 1.27e11·31-s + 3.43e10·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.0579·3-s + 0.5·4-s − 1.42·5-s − 0.0410·6-s + 0.353·8-s − 0.996·9-s − 1.01·10-s + 0.512·11-s − 0.0289·12-s − 0.224·13-s + 0.0828·15-s + 0.250·16-s + 1.79·17-s − 0.704·18-s + 1.41·19-s − 0.714·20-s + 0.362·22-s + 0.195·23-s − 0.0205·24-s + 1.04·25-s − 0.158·26-s + 0.115·27-s − 1.69·29-s + 0.0585·30-s + 0.835·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - 128T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 219.T + 1.43e7T^{2} \) |
| 5 | \( 1 + 2.49e5T + 3.05e10T^{2} \) |
| 11 | \( 1 - 3.30e7T + 4.17e15T^{2} \) |
| 13 | \( 1 + 5.07e7T + 5.11e16T^{2} \) |
| 17 | \( 1 - 3.03e9T + 2.86e18T^{2} \) |
| 19 | \( 1 - 5.51e9T + 1.51e19T^{2} \) |
| 23 | \( 1 - 3.19e9T + 2.66e20T^{2} \) |
| 29 | \( 1 + 1.57e11T + 8.62e21T^{2} \) |
| 31 | \( 1 - 1.27e11T + 2.34e22T^{2} \) |
| 37 | \( 1 - 2.62e10T + 3.33e23T^{2} \) |
| 41 | \( 1 - 1.21e11T + 1.55e24T^{2} \) |
| 43 | \( 1 + 2.25e12T + 3.17e24T^{2} \) |
| 47 | \( 1 - 1.69e12T + 1.20e25T^{2} \) |
| 53 | \( 1 + 4.70e12T + 7.31e25T^{2} \) |
| 59 | \( 1 + 2.80e13T + 3.65e26T^{2} \) |
| 61 | \( 1 + 3.64e13T + 6.02e26T^{2} \) |
| 67 | \( 1 + 9.23e13T + 2.46e27T^{2} \) |
| 71 | \( 1 - 5.57e13T + 5.87e27T^{2} \) |
| 73 | \( 1 + 4.08e13T + 8.90e27T^{2} \) |
| 79 | \( 1 - 2.39e14T + 2.91e28T^{2} \) |
| 83 | \( 1 - 1.19e14T + 6.11e28T^{2} \) |
| 89 | \( 1 + 1.40e14T + 1.74e29T^{2} \) |
| 97 | \( 1 - 4.92e14T + 6.33e29T^{2} \) |
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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
−10.91345503895700229996966208347, −9.430446671281170498096413837964, −8.001631929585295907947650920791, −7.38712293654114335716158933675, −5.91926730129659162106740893711, −4.91130146870919817939161227806, −3.60075703331006433138077097828, −3.06864457752078324070391132912, −1.23538225355236515878559192442, 0,
1.23538225355236515878559192442, 3.06864457752078324070391132912, 3.60075703331006433138077097828, 4.91130146870919817939161227806, 5.91926730129659162106740893711, 7.38712293654114335716158933675, 8.001631929585295907947650920791, 9.430446671281170498096413837964, 10.91345503895700229996966208347
