| L(s) = 1 | − 6.85·2-s + 15.3·3-s − 80.9·4-s − 335.·5-s − 105.·6-s + 735.·7-s + 1.43e3·8-s − 1.95e3·9-s + 2.30e3·10-s + 1.52e3·11-s − 1.24e3·12-s − 826.·13-s − 5.04e3·14-s − 5.15e3·15-s + 534.·16-s + 3.52e4·17-s + 1.33e4·18-s + 3.67e4·19-s + 2.71e4·20-s + 1.12e4·21-s − 1.04e4·22-s + 1.85e4·23-s + 2.20e4·24-s + 3.43e4·25-s + 5.66e3·26-s − 6.35e4·27-s − 5.95e4·28-s + ⋯ |
| L(s) = 1 | − 0.606·2-s + 0.328·3-s − 0.632·4-s − 1.20·5-s − 0.199·6-s + 0.810·7-s + 0.989·8-s − 0.892·9-s + 0.727·10-s + 0.344·11-s − 0.207·12-s − 0.104·13-s − 0.491·14-s − 0.394·15-s + 0.0326·16-s + 1.73·17-s + 0.540·18-s + 1.23·19-s + 0.759·20-s + 0.266·21-s − 0.208·22-s + 0.317·23-s + 0.325·24-s + 0.440·25-s + 0.0632·26-s − 0.621·27-s − 0.512·28-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
| \(L(4)\) |
\(\approx\) |
\(0.9919286529\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9919286529\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + 6.85T + 128T^{2} \) |
| 3 | \( 1 - 15.3T + 2.18e3T^{2} \) |
| 5 | \( 1 + 335.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 735.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 1.52e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 826.T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.52e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.67e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.85e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.35e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.42e4T + 2.75e10T^{2} \) |
| 41 | \( 1 - 6.07e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 7.01e3T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.04e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 7.18e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 7.67e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.65e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.38e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 5.19e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.55e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.84e5T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.25e5T + 2.71e13T^{2} \) |
| 89 | \( 1 - 9.98e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 9.16e6T + 8.07e13T^{2} \) |
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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
−14.66553537882045623642009777959, −14.00417971228825710976926585173, −12.19101664488537597988044334495, −11.19043953235607414538476369858, −9.602303095442423892755521144700, −8.266336890785673338045023892742, −7.69490308489381776256491127393, −5.10647504780188132743307517862, −3.52108124210253391847801265695, −0.876447989551867241953528249837,
0.876447989551867241953528249837, 3.52108124210253391847801265695, 5.10647504780188132743307517862, 7.69490308489381776256491127393, 8.266336890785673338045023892742, 9.602303095442423892755521144700, 11.19043953235607414538476369858, 12.19101664488537597988044334495, 14.00417971228825710976926585173, 14.66553537882045623642009777959
