| L(s) = 1 | − 2.69e3·3-s − 1.60e5·5-s + 2.45e6·7-s − 7.09e6·9-s − 2.90e6·11-s + 3.40e8·13-s + 4.32e8·15-s + 2.60e9·17-s + 8.93e8·19-s − 6.61e9·21-s + 2.47e10·23-s − 4.79e9·25-s + 5.77e10·27-s − 6.85e10·29-s − 1.50e11·31-s + 7.83e9·33-s − 3.93e11·35-s + 7.16e11·37-s − 9.16e11·39-s + 8.93e11·41-s + 1.21e11·43-s + 1.13e12·45-s + 5.62e12·47-s + 1.28e12·49-s − 7.01e12·51-s − 1.44e13·53-s + 4.66e11·55-s + ⋯ |
| L(s) = 1 | − 0.711·3-s − 0.918·5-s + 1.12·7-s − 0.494·9-s − 0.0449·11-s + 1.50·13-s + 0.652·15-s + 1.53·17-s + 0.229·19-s − 0.801·21-s + 1.51·23-s − 0.157·25-s + 1.06·27-s − 0.737·29-s − 0.983·31-s + 0.0319·33-s − 1.03·35-s + 1.24·37-s − 1.06·39-s + 0.716·41-s + 0.0678·43-s + 0.453·45-s + 1.61·47-s + 0.270·49-s − 1.09·51-s − 1.68·53-s + 0.0413·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(1.890812352\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.890812352\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 8.93e8T \) |
| good | 3 | \( 1 + 2.69e3T + 1.43e7T^{2} \) |
| 5 | \( 1 + 1.60e5T + 3.05e10T^{2} \) |
| 7 | \( 1 - 2.45e6T + 4.74e12T^{2} \) |
| 11 | \( 1 + 2.90e6T + 4.17e15T^{2} \) |
| 13 | \( 1 - 3.40e8T + 5.11e16T^{2} \) |
| 17 | \( 1 - 2.60e9T + 2.86e18T^{2} \) |
| 23 | \( 1 - 2.47e10T + 2.66e20T^{2} \) |
| 29 | \( 1 + 6.85e10T + 8.62e21T^{2} \) |
| 31 | \( 1 + 1.50e11T + 2.34e22T^{2} \) |
| 37 | \( 1 - 7.16e11T + 3.33e23T^{2} \) |
| 41 | \( 1 - 8.93e11T + 1.55e24T^{2} \) |
| 43 | \( 1 - 1.21e11T + 3.17e24T^{2} \) |
| 47 | \( 1 - 5.62e12T + 1.20e25T^{2} \) |
| 53 | \( 1 + 1.44e13T + 7.31e25T^{2} \) |
| 59 | \( 1 - 1.37e13T + 3.65e26T^{2} \) |
| 61 | \( 1 + 3.98e12T + 6.02e26T^{2} \) |
| 67 | \( 1 - 2.07e12T + 2.46e27T^{2} \) |
| 71 | \( 1 + 2.83e13T + 5.87e27T^{2} \) |
| 73 | \( 1 + 1.21e14T + 8.90e27T^{2} \) |
| 79 | \( 1 + 2.89e13T + 2.91e28T^{2} \) |
| 83 | \( 1 - 1.12e13T + 6.11e28T^{2} \) |
| 89 | \( 1 - 5.05e14T + 1.74e29T^{2} \) |
| 97 | \( 1 + 6.41e14T + 6.33e29T^{2} \) |
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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
−10.79054121822170866631199898895, −9.127610953328459627607756410455, −8.127496351122594011190027574600, −7.43499733663371389133270747675, −5.96369644622420442098835201382, −5.24044773491630508619420128137, −4.08010373247863016700572698447, −3.08209846992882988398134057441, −1.39087679781249238486741904093, −0.66406514245009023243463219604,
0.66406514245009023243463219604, 1.39087679781249238486741904093, 3.08209846992882988398134057441, 4.08010373247863016700572698447, 5.24044773491630508619420128137, 5.96369644622420442098835201382, 7.43499733663371389133270747675, 8.127496351122594011190027574600, 9.127610953328459627607756410455, 10.79054121822170866631199898895
