| L(s) = 1 | − 8·2-s − 86.3·3-s + 64·4-s − 4.83·5-s + 691.·6-s − 1.23e3·7-s − 512·8-s + 5.27e3·9-s + 38.6·10-s − 2.29e3·11-s − 5.52e3·12-s − 6.23e3·13-s + 9.87e3·14-s + 417.·15-s + 4.09e3·16-s + 4.48e3·17-s − 4.22e4·18-s − 1.12e4·19-s − 309.·20-s + 1.06e5·21-s + 1.83e4·22-s + 6.00e4·23-s + 4.42e4·24-s − 7.81e4·25-s + 4.98e4·26-s − 2.66e5·27-s − 7.90e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.84·3-s + 0.5·4-s − 0.0172·5-s + 1.30·6-s − 1.36·7-s − 0.353·8-s + 2.41·9-s + 0.0122·10-s − 0.520·11-s − 0.923·12-s − 0.786·13-s + 0.961·14-s + 0.0319·15-s + 0.250·16-s + 0.221·17-s − 1.70·18-s − 0.375·19-s − 0.00864·20-s + 2.51·21-s + 0.367·22-s + 1.02·23-s + 0.653·24-s − 0.999·25-s + 0.556·26-s − 2.60·27-s − 0.680·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0003930892459\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0003930892459\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8T \) |
| 269 | \( 1 + 1.94e7T \) |
| good | 3 | \( 1 + 86.3T + 2.18e3T^{2} \) |
| 5 | \( 1 + 4.83T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.23e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 2.29e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 6.23e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 4.48e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.12e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 6.00e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.76e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.04e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.83e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 7.55e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.37e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.21e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 3.79e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 3.24e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.55e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.52e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 5.60e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 5.79e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 6.48e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 4.69e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.02e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 5.04e6T + 8.07e13T^{2} \) |
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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
−10.08314569752242470474043924274, −9.106522143266597436040771665486, −7.56572110094194766778424663484, −6.91679915063996029199196590254, −6.08673457023407909021041057301, −5.41512716139458920547621589343, −4.23273676966403631412037867182, −2.79585078923602657906985556277, −1.32047705280584305721065365342, −0.01132391058577910467166311505,
0.01132391058577910467166311505, 1.32047705280584305721065365342, 2.79585078923602657906985556277, 4.23273676966403631412037867182, 5.41512716139458920547621589343, 6.08673457023407909021041057301, 6.91679915063996029199196590254, 7.56572110094194766778424663484, 9.106522143266597436040771665486, 10.08314569752242470474043924274
