| L(s) = 1 | − 16.0·2-s + 130.·4-s − 431.·5-s + 218.·7-s − 36.2·8-s + 6.94e3·10-s + 7.01e3·11-s + 7.01e3·13-s − 3.51e3·14-s − 1.60e4·16-s − 3.70e4·17-s − 3.98e4·19-s − 5.62e4·20-s − 1.12e5·22-s − 2.98e4·23-s + 1.08e5·25-s − 1.12e5·26-s + 2.85e4·28-s + 7.28e4·29-s − 1.03e5·31-s + 2.63e5·32-s + 5.95e5·34-s − 9.45e4·35-s + 3.30e5·37-s + 6.40e5·38-s + 1.56e4·40-s − 3.87e5·41-s + ⋯ |
| L(s) = 1 | − 1.42·2-s + 1.01·4-s − 1.54·5-s + 0.241·7-s − 0.0250·8-s + 2.19·10-s + 1.58·11-s + 0.885·13-s − 0.342·14-s − 0.982·16-s − 1.82·17-s − 1.33·19-s − 1.57·20-s − 2.25·22-s − 0.511·23-s + 1.38·25-s − 1.25·26-s + 0.245·28-s + 0.554·29-s − 0.621·31-s + 1.41·32-s + 2.59·34-s − 0.372·35-s + 1.07·37-s + 1.89·38-s + 0.0386·40-s − 0.879·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 + 7.95e4T \) |
| good | 2 | \( 1 + 16.0T + 128T^{2} \) |
| 5 | \( 1 + 431.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 218.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 7.01e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 7.01e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.70e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.98e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 2.98e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 7.28e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.03e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.30e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.87e5T + 1.94e11T^{2} \) |
| 47 | \( 1 - 1.39e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 3.00e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 4.40e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.72e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.45e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.85e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 6.73e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 5.44e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 4.84e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 8.98e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 7.64e6T + 8.07e13T^{2} \) |
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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
−9.392586837980374967794910397585, −8.556439645068259344216435468485, −8.285054602240937529306785849716, −7.03395076216842238148136567141, −6.47231668831053172644762937955, −4.35372717016805692496912362971, −3.92594761158582707825823056505, −2.09023473161212212623582314343, −0.924867206244969590485219580196, 0,
0.924867206244969590485219580196, 2.09023473161212212623582314343, 3.92594761158582707825823056505, 4.35372717016805692496912362971, 6.47231668831053172644762937955, 7.03395076216842238148136567141, 8.285054602240937529306785849716, 8.556439645068259344216435468485, 9.392586837980374967794910397585
