| L(s) = 1 | − 128·2-s + 3.34e3·3-s + 1.63e4·4-s − 2.59e5·5-s − 4.28e5·6-s − 2.09e6·8-s − 3.14e6·9-s + 3.31e7·10-s − 2.28e7·11-s + 5.48e7·12-s + 4.48e8·13-s − 8.68e8·15-s + 2.68e8·16-s + 6.06e8·17-s + 4.02e8·18-s + 3.59e9·19-s − 4.24e9·20-s + 2.92e9·22-s − 2.10e10·23-s − 7.01e9·24-s + 3.67e10·25-s − 5.73e10·26-s − 5.85e10·27-s − 1.19e11·29-s + 1.11e11·30-s − 1.18e11·31-s − 3.43e10·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.883·3-s + 0.5·4-s − 1.48·5-s − 0.624·6-s − 0.353·8-s − 0.219·9-s + 1.04·10-s − 0.353·11-s + 0.441·12-s + 1.98·13-s − 1.31·15-s + 0.250·16-s + 0.358·17-s + 0.155·18-s + 0.923·19-s − 0.742·20-s + 0.249·22-s − 1.29·23-s − 0.312·24-s + 1.20·25-s − 1.40·26-s − 1.07·27-s − 1.28·29-s + 0.927·30-s − 0.772·31-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
| \(L(8)\) |
\(\approx\) |
\(1.199437649\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.199437649\) |
| \(L(\frac{17}{2})\) |
|
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 | \( 1 - 3.34e3T + 1.43e7T^{2} \) |
| 5 | \( 1 + 2.59e5T + 3.05e10T^{2} \) |
| 11 | \( 1 + 2.28e7T + 4.17e15T^{2} \) |
| 13 | \( 1 - 4.48e8T + 5.11e16T^{2} \) |
| 17 | \( 1 - 6.06e8T + 2.86e18T^{2} \) |
| 19 | \( 1 - 3.59e9T + 1.51e19T^{2} \) |
| 23 | \( 1 + 2.10e10T + 2.66e20T^{2} \) |
| 29 | \( 1 + 1.19e11T + 8.62e21T^{2} \) |
| 31 | \( 1 + 1.18e11T + 2.34e22T^{2} \) |
| 37 | \( 1 - 2.59e11T + 3.33e23T^{2} \) |
| 41 | \( 1 + 1.79e12T + 1.55e24T^{2} \) |
| 43 | \( 1 + 2.09e12T + 3.17e24T^{2} \) |
| 47 | \( 1 + 6.46e11T + 1.20e25T^{2} \) |
| 53 | \( 1 + 7.11e11T + 7.31e25T^{2} \) |
| 59 | \( 1 - 2.75e13T + 3.65e26T^{2} \) |
| 61 | \( 1 - 2.31e13T + 6.02e26T^{2} \) |
| 67 | \( 1 - 6.37e13T + 2.46e27T^{2} \) |
| 71 | \( 1 - 1.52e13T + 5.87e27T^{2} \) |
| 73 | \( 1 - 7.48e13T + 8.90e27T^{2} \) |
| 79 | \( 1 + 8.63e12T + 2.91e28T^{2} \) |
| 83 | \( 1 + 3.87e14T + 6.11e28T^{2} \) |
| 89 | \( 1 + 5.90e14T + 1.74e29T^{2} \) |
| 97 | \( 1 + 5.78e14T + 6.33e29T^{2} \) |
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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
−11.18374439761518187781735375235, −9.781862644196184724987091420965, −8.455220705435769320721677529120, −8.220578748855733169113945258223, −7.18492819018894198179359420496, −5.68546751000962963578697388985, −3.74886707867745157834264106987, −3.36278680570438589138758721896, −1.81535354291424934954454330842, −0.52100404499468618706105478948,
0.52100404499468618706105478948, 1.81535354291424934954454330842, 3.36278680570438589138758721896, 3.74886707867745157834264106987, 5.68546751000962963578697388985, 7.18492819018894198179359420496, 8.220578748855733169113945258223, 8.455220705435769320721677529120, 9.781862644196184724987091420965, 11.18374439761518187781735375235
