| L(s) = 1 | − 8·2-s − 53.1·3-s + 64·4-s − 125·5-s + 425.·6-s + 1.53e3·7-s − 512·8-s + 641.·9-s + 1.00e3·10-s − 5.52e3·11-s − 3.40e3·12-s + 1.33e4·13-s − 1.23e4·14-s + 6.64e3·15-s + 4.09e3·16-s − 1.29e4·17-s − 5.12e3·18-s − 3.25e4·19-s − 8.00e3·20-s − 8.18e4·21-s + 4.41e4·22-s − 1.58e4·23-s + 2.72e4·24-s + 1.56e4·25-s − 1.07e5·26-s + 8.22e4·27-s + 9.85e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.13·3-s + 0.5·4-s − 0.447·5-s + 0.804·6-s + 1.69·7-s − 0.353·8-s + 0.293·9-s + 0.316·10-s − 1.25·11-s − 0.568·12-s + 1.68·13-s − 1.19·14-s + 0.508·15-s + 0.250·16-s − 0.641·17-s − 0.207·18-s − 1.08·19-s − 0.223·20-s − 1.92·21-s + 0.884·22-s − 0.271·23-s + 0.402·24-s + 0.199·25-s − 1.19·26-s + 0.803·27-s + 0.848·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + 8T \) |
| 5 | \( 1 + 125T \) |
| 31 | \( 1 - 2.97e4T \) |
| good | 3 | \( 1 + 53.1T + 2.18e3T^{2} \) |
| 7 | \( 1 - 1.53e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 5.52e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.33e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.29e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.25e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.58e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 7.06e4T + 1.72e10T^{2} \) |
| 37 | \( 1 + 5.39e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.22e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 5.73e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 9.29e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 6.56e4T + 1.17e12T^{2} \) |
| 59 | \( 1 - 9.74e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.85e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.16e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.10e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.05e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.69e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 7.41e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 5.80e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.48e7T + 8.07e13T^{2} \) |
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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
−10.63863337392573531521837488156, −8.679996656666741462816159882719, −8.293660570777776339708564508287, −7.22515773722469750211319590445, −6.00740614946798795977013860763, −5.19885477851822604934628034207, −4.08862196297858400313116921841, −2.23274030983479114006613027380, −1.06699615105444387296562391823, 0,
1.06699615105444387296562391823, 2.23274030983479114006613027380, 4.08862196297858400313116921841, 5.19885477851822604934628034207, 6.00740614946798795977013860763, 7.22515773722469750211319590445, 8.293660570777776339708564508287, 8.679996656666741462816159882719, 10.63863337392573531521837488156
