| L(s) = 1 | + 17.6·2-s + 0.554·3-s + 185.·4-s + 268.·5-s + 9.81·6-s + 1.09e3·7-s + 1.01e3·8-s − 2.18e3·9-s + 4.75e3·10-s + 651.·11-s + 102.·12-s + 408.·13-s + 1.93e4·14-s + 148.·15-s − 5.80e3·16-s + 2.61e4·17-s − 3.86e4·18-s + 5.65e4·19-s + 4.97e4·20-s + 605.·21-s + 1.15e4·22-s − 5.27e4·23-s + 561.·24-s − 6.05e3·25-s + 7.22e3·26-s − 2.42e3·27-s + 2.02e5·28-s + ⋯ |
| L(s) = 1 | + 1.56·2-s + 0.0118·3-s + 1.44·4-s + 0.960·5-s + 0.0185·6-s + 1.20·7-s + 0.698·8-s − 0.999·9-s + 1.50·10-s + 0.147·11-s + 0.0171·12-s + 0.0515·13-s + 1.88·14-s + 0.0113·15-s − 0.354·16-s + 1.28·17-s − 1.56·18-s + 1.89·19-s + 1.38·20-s + 0.0142·21-s + 0.230·22-s − 0.903·23-s + 0.00828·24-s − 0.0775·25-s + 0.0805·26-s − 0.0237·27-s + 1.73·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(5.087153264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.087153264\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 47 | \( 1 + 1.03e5T \) |
| good | 2 | \( 1 - 17.6T + 128T^{2} \) |
| 3 | \( 1 - 0.554T + 2.18e3T^{2} \) |
| 5 | \( 1 - 268.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.09e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 651.T + 1.94e7T^{2} \) |
| 13 | \( 1 - 408.T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.61e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 5.65e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 5.27e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.48e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.05e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.18e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 1.14e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 8.35e4T + 2.71e11T^{2} \) |
| 53 | \( 1 + 1.61e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 8.85e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 8.57e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.74e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.07e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.18e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.64e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.58e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.05e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 3.90e6T + 8.07e13T^{2} \) |
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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
−14.32998767035225032027407080864, −13.37748041262416241715804847196, −11.97621829183960340946226098452, −11.22020588820222799714847173058, −9.468834167710872497052005409826, −7.68831907990727353756762675114, −5.76130267885195944040122032649, −5.28437372105912527511195785280, −3.44676674995659927525223330051, −1.84862854993962179194203577397,
1.84862854993962179194203577397, 3.44676674995659927525223330051, 5.28437372105912527511195785280, 5.76130267885195944040122032649, 7.68831907990727353756762675114, 9.468834167710872497052005409826, 11.22020588820222799714847173058, 11.97621829183960340946226098452, 13.37748041262416241715804847196, 14.32998767035225032027407080864
