| L(s) = 1 | − 3.12·2-s − 59.7·3-s − 118.·4-s − 342.·5-s + 186.·6-s − 975.·7-s + 768.·8-s + 1.38e3·9-s + 1.06e3·10-s − 2.24e3·11-s + 7.06e3·12-s − 1.52e3·13-s + 3.04e3·14-s + 2.04e4·15-s + 1.27e4·16-s − 1.59e4·17-s − 4.33e3·18-s − 1.39e4·19-s + 4.05e4·20-s + 5.83e4·21-s + 6.99e3·22-s − 6.22e3·23-s − 4.59e4·24-s + 3.93e4·25-s + 4.77e3·26-s + 4.78e4·27-s + 1.15e5·28-s + ⋯ |
| L(s) = 1 | − 0.275·2-s − 1.27·3-s − 0.923·4-s − 1.22·5-s + 0.352·6-s − 1.07·7-s + 0.530·8-s + 0.634·9-s + 0.338·10-s − 0.507·11-s + 1.18·12-s − 0.193·13-s + 0.296·14-s + 1.56·15-s + 0.777·16-s − 0.784·17-s − 0.175·18-s − 0.466·19-s + 1.13·20-s + 1.37·21-s + 0.140·22-s − 0.106·23-s − 0.678·24-s + 0.503·25-s + 0.0532·26-s + 0.467·27-s + 0.992·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.06597448211\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06597448211\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + 5.06e4T \) |
| good | 2 | \( 1 + 3.12T + 128T^{2} \) |
| 3 | \( 1 + 59.7T + 2.18e3T^{2} \) |
| 5 | \( 1 + 342.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 975.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 2.24e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.52e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.59e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.39e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 6.22e3T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.31e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.00e5T + 2.75e10T^{2} \) |
| 41 | \( 1 + 8.10e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 7.97e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 7.88e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 3.99e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.66e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 3.09e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.43e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.16e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.63e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.17e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 4.24e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + 6.23e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.61e7T + 8.07e13T^{2} \) |
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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
−15.17963022225854508947700069352, −13.29581158783132895204391334046, −12.36105007487935761079339329807, −11.19148263206075844913660205952, −10.00233651285203417608815647320, −8.468660769119640462099057291368, −6.90994796560625555826098108482, −5.26899412429609551721775346666, −3.84552991494495652333892676751, −0.21130406350845247512526609604,
0.21130406350845247512526609604, 3.84552991494495652333892676751, 5.26899412429609551721775346666, 6.90994796560625555826098108482, 8.468660769119640462099057291368, 10.00233651285203417608815647320, 11.19148263206075844913660205952, 12.36105007487935761079339329807, 13.29581158783132895204391334046, 15.17963022225854508947700069352
