| L(s) = 1 | + 60.5·2-s − 243·3-s + 1.62e3·4-s + 3.73e3·5-s − 1.47e4·6-s + 5.15e4·7-s − 2.58e4·8-s + 5.90e4·9-s + 2.26e5·10-s + 8.19e3·11-s − 3.93e5·12-s − 2.10e6·13-s + 3.12e6·14-s − 9.07e5·15-s − 4.88e6·16-s + 4.38e6·17-s + 3.57e6·18-s − 3.78e6·19-s + 6.05e6·20-s − 1.25e7·21-s + 4.96e5·22-s + 2.08e7·23-s + 6.28e6·24-s − 3.48e7·25-s − 1.27e8·26-s − 1.43e7·27-s + 8.35e7·28-s + ⋯ |
| L(s) = 1 | + 1.33·2-s − 0.577·3-s + 0.791·4-s + 0.534·5-s − 0.772·6-s + 1.15·7-s − 0.279·8-s + 0.333·9-s + 0.715·10-s + 0.0153·11-s − 0.456·12-s − 1.56·13-s + 1.55·14-s − 0.308·15-s − 1.16·16-s + 0.749·17-s + 0.446·18-s − 0.351·19-s + 0.422·20-s − 0.669·21-s + 0.0205·22-s + 0.676·23-s + 0.161·24-s − 0.714·25-s − 2.10·26-s − 0.192·27-s + 0.917·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 243T \) |
| 59 | \( 1 - 7.14e8T \) |
| good | 2 | \( 1 - 60.5T + 2.04e3T^{2} \) |
| 5 | \( 1 - 3.73e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 5.15e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 8.19e3T + 2.85e11T^{2} \) |
| 13 | \( 1 + 2.10e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 4.38e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 3.78e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 2.08e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.72e8T + 1.22e16T^{2} \) |
| 31 | \( 1 - 2.05e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 4.55e7T + 1.77e17T^{2} \) |
| 41 | \( 1 + 1.14e9T + 5.50e17T^{2} \) |
| 43 | \( 1 - 6.27e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 1.39e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 5.76e9T + 9.26e18T^{2} \) |
| 61 | \( 1 + 9.92e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 2.96e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 3.75e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + 1.30e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 1.07e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 3.38e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 7.13e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 5.74e10T + 7.15e21T^{2} \) |
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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
−10.39785737248621836005615770562, −9.340402814772553471699365529999, −7.85585320620936247781722286938, −6.75030450999553987419907841532, −5.51113434880213799927948839913, −5.07124916326801583088046527422, −4.10593209374431261898965247147, −2.65436443240584474947205936577, −1.59578672301398120187227618343, 0,
1.59578672301398120187227618343, 2.65436443240584474947205936577, 4.10593209374431261898965247147, 5.07124916326801583088046527422, 5.51113434880213799927948839913, 6.75030450999553987419907841532, 7.85585320620936247781722286938, 9.340402814772553471699365529999, 10.39785737248621836005615770562
