| L(s) = 1 | + 19.7·2-s + 5.55·3-s + 261.·4-s − 428.·5-s + 109.·6-s − 343·7-s + 2.62e3·8-s − 2.15e3·9-s − 8.45e3·10-s − 6.10e3·11-s + 1.45e3·12-s + 2.19e3·13-s − 6.76e3·14-s − 2.37e3·15-s + 1.84e4·16-s − 2.31e3·17-s − 4.25e4·18-s + 2.85e4·19-s − 1.11e5·20-s − 1.90e3·21-s − 1.20e5·22-s − 6.48e4·23-s + 1.45e4·24-s + 1.05e5·25-s + 4.33e4·26-s − 2.41e4·27-s − 8.95e4·28-s + ⋯ |
| L(s) = 1 | + 1.74·2-s + 0.118·3-s + 2.04·4-s − 1.53·5-s + 0.207·6-s − 0.377·7-s + 1.81·8-s − 0.985·9-s − 2.67·10-s − 1.38·11-s + 0.242·12-s + 0.277·13-s − 0.659·14-s − 0.182·15-s + 1.12·16-s − 0.114·17-s − 1.71·18-s + 0.956·19-s − 3.12·20-s − 0.0448·21-s − 2.41·22-s − 1.11·23-s + 0.215·24-s + 1.35·25-s + 0.483·26-s − 0.235·27-s − 0.771·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 + 343T \) |
| 13 | \( 1 - 2.19e3T \) |
| good | 2 | \( 1 - 19.7T + 128T^{2} \) |
| 3 | \( 1 - 5.55T + 2.18e3T^{2} \) |
| 5 | \( 1 + 428.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 6.10e3T + 1.94e7T^{2} \) |
| 17 | \( 1 + 2.31e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.85e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 6.48e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.53e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.75e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 8.94e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 5.67e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.98e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 9.22e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 7.93e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.62e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 3.68e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.48e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.76e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.08e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 5.15e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 3.78e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.12e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.55e7T + 8.07e13T^{2} \) |
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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
−12.07750894542445953315905138543, −11.67984252086260510523263391033, −10.50600202115378974634516614902, −8.339851420813811705915022060470, −7.37079941071196105835014686134, −5.93906198508755219794409312433, −4.79673481247408652321443049339, −3.57038044990405343388975565955, −2.74620011669511967285979721546, 0,
2.74620011669511967285979721546, 3.57038044990405343388975565955, 4.79673481247408652321443049339, 5.93906198508755219794409312433, 7.37079941071196105835014686134, 8.339851420813811705915022060470, 10.50600202115378974634516614902, 11.67984252086260510523263391033, 12.07750894542445953315905138543
