| L(s) = 1 | − 125.·2-s + 4.14e3·3-s − 1.69e4·4-s − 7.81e4·5-s − 5.20e5·6-s + 4.19e6·7-s + 6.25e6·8-s + 2.84e6·9-s + 9.81e6·10-s + 5.50e7·11-s − 7.04e7·12-s + 3.01e8·13-s − 5.27e8·14-s − 3.23e8·15-s − 2.28e8·16-s − 2.80e8·17-s − 3.57e8·18-s − 4.05e9·19-s + 1.32e9·20-s + 1.74e10·21-s − 6.91e9·22-s + 4.20e9·23-s + 2.59e10·24-s + 6.10e9·25-s − 3.79e10·26-s − 4.76e10·27-s − 7.13e10·28-s + ⋯ |
| L(s) = 1 | − 0.693·2-s + 1.09·3-s − 0.518·4-s − 0.447·5-s − 0.759·6-s + 1.92·7-s + 1.05·8-s + 0.198·9-s + 0.310·10-s + 0.852·11-s − 0.567·12-s + 1.33·13-s − 1.33·14-s − 0.489·15-s − 0.212·16-s − 0.165·17-s − 0.137·18-s − 1.04·19-s + 0.231·20-s + 2.10·21-s − 0.591·22-s + 0.257·23-s + 1.15·24-s + 0.200·25-s − 0.926·26-s − 0.877·27-s − 0.998·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(1.621723312\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.621723312\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 7.81e4T \) |
| good | 2 | \( 1 + 125.T + 3.27e4T^{2} \) |
| 3 | \( 1 - 4.14e3T + 1.43e7T^{2} \) |
| 7 | \( 1 - 4.19e6T + 4.74e12T^{2} \) |
| 11 | \( 1 - 5.50e7T + 4.17e15T^{2} \) |
| 13 | \( 1 - 3.01e8T + 5.11e16T^{2} \) |
| 17 | \( 1 + 2.80e8T + 2.86e18T^{2} \) |
| 19 | \( 1 + 4.05e9T + 1.51e19T^{2} \) |
| 23 | \( 1 - 4.20e9T + 2.66e20T^{2} \) |
| 29 | \( 1 - 1.40e9T + 8.62e21T^{2} \) |
| 31 | \( 1 - 1.19e11T + 2.34e22T^{2} \) |
| 37 | \( 1 - 3.56e11T + 3.33e23T^{2} \) |
| 41 | \( 1 + 5.16e11T + 1.55e24T^{2} \) |
| 43 | \( 1 - 5.03e11T + 3.17e24T^{2} \) |
| 47 | \( 1 + 4.20e12T + 1.20e25T^{2} \) |
| 53 | \( 1 - 6.08e12T + 7.31e25T^{2} \) |
| 59 | \( 1 + 3.64e13T + 3.65e26T^{2} \) |
| 61 | \( 1 + 1.45e13T + 6.02e26T^{2} \) |
| 67 | \( 1 - 6.54e13T + 2.46e27T^{2} \) |
| 71 | \( 1 + 2.67e13T + 5.87e27T^{2} \) |
| 73 | \( 1 + 5.52e13T + 8.90e27T^{2} \) |
| 79 | \( 1 + 6.24e13T + 2.91e28T^{2} \) |
| 83 | \( 1 + 5.20e13T + 6.11e28T^{2} \) |
| 89 | \( 1 + 5.44e14T + 1.74e29T^{2} \) |
| 97 | \( 1 + 1.36e14T + 6.33e29T^{2} \) |
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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
−19.92220269449593080105437921672, −18.56309795129889657878432284442, −17.24297316014388840182692601711, −14.87998187206835030630311047168, −13.79863365043148166334423495598, −11.13985507916071075878944969507, −8.776147352651220105004617336147, −8.083797284083366139761820868218, −4.24057782939355979769416371380, −1.45402524457395868057460669727,
1.45402524457395868057460669727, 4.24057782939355979769416371380, 8.083797284083366139761820868218, 8.776147352651220105004617336147, 11.13985507916071075878944969507, 13.79863365043148166334423495598, 14.87998187206835030630311047168, 17.24297316014388840182692601711, 18.56309795129889657878432284442, 19.92220269449593080105437921672
