| L(s) = 1 | + 8.49·2-s + 45.4·3-s − 55.8·4-s − 130.·5-s + 386.·6-s + 343·7-s − 1.56e3·8-s − 120.·9-s − 1.10e3·10-s + 1.04e3·11-s − 2.53e3·12-s − 2.19e3·13-s + 2.91e3·14-s − 5.92e3·15-s − 6.11e3·16-s − 2.97e4·17-s − 1.02e3·18-s + 5.79e3·19-s + 7.28e3·20-s + 1.55e4·21-s + 8.85e3·22-s − 9.21e4·23-s − 7.09e4·24-s − 6.11e4·25-s − 1.86e4·26-s − 1.04e5·27-s − 1.91e4·28-s + ⋯ |
| L(s) = 1 | + 0.750·2-s + 0.972·3-s − 0.436·4-s − 0.466·5-s + 0.729·6-s + 0.377·7-s − 1.07·8-s − 0.0549·9-s − 0.350·10-s + 0.236·11-s − 0.424·12-s − 0.277·13-s + 0.283·14-s − 0.453·15-s − 0.373·16-s − 1.46·17-s − 0.0412·18-s + 0.193·19-s + 0.203·20-s + 0.367·21-s + 0.177·22-s − 1.57·23-s − 1.04·24-s − 0.782·25-s − 0.208·26-s − 1.02·27-s − 0.164·28-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 | \( 1 - 8.49T + 128T^{2} \) |
| 3 | \( 1 - 45.4T + 2.18e3T^{2} \) |
| 5 | \( 1 + 130.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 1.04e3T + 1.94e7T^{2} \) |
| 17 | \( 1 + 2.97e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 5.79e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 9.21e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 3.48e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.86e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.84e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 9.96e3T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.36e4T + 2.71e11T^{2} \) |
| 47 | \( 1 - 4.09e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.24e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.59e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.14e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.72e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.80e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 6.02e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.01e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.81e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 4.78e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 4.85e6T + 8.07e13T^{2} \) |
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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
−12.30885995737762512855284699599, −11.37669300301521540464917215795, −9.685125782505193797727268422148, −8.691501629476925124243564714705, −7.83721338322399309573901049107, −6.12791026850160341885994766130, −4.59741072238162505054265542284, −3.66679164738540748871247680094, −2.29942721829494644228638490171, 0,
2.29942721829494644228638490171, 3.66679164738540748871247680094, 4.59741072238162505054265542284, 6.12791026850160341885994766130, 7.83721338322399309573901049107, 8.691501629476925124243564714705, 9.685125782505193797727268422148, 11.37669300301521540464917215795, 12.30885995737762512855284699599
