| L(s) = 1 | − 26.3·2-s − 243·3-s − 1.35e3·4-s + 1.24e4·5-s + 6.39e3·6-s − 4.83e4·7-s + 8.96e4·8-s + 5.90e4·9-s − 3.28e5·10-s + 5.24e5·11-s + 3.29e5·12-s − 3.61e4·13-s + 1.27e6·14-s − 3.03e6·15-s + 4.13e5·16-s + 2.33e6·17-s − 1.55e6·18-s − 1.72e7·19-s − 1.69e7·20-s + 1.17e7·21-s − 1.38e7·22-s − 3.09e7·23-s − 2.17e7·24-s + 1.07e8·25-s + 9.51e5·26-s − 1.43e7·27-s + 6.55e7·28-s + ⋯ |
| L(s) = 1 | − 0.581·2-s − 0.577·3-s − 0.661·4-s + 1.78·5-s + 0.335·6-s − 1.08·7-s + 0.966·8-s + 0.333·9-s − 1.03·10-s + 0.982·11-s + 0.381·12-s − 0.0269·13-s + 0.633·14-s − 1.03·15-s + 0.0986·16-s + 0.399·17-s − 0.193·18-s − 1.59·19-s − 1.18·20-s + 0.628·21-s − 0.571·22-s − 1.00·23-s − 0.558·24-s + 2.19·25-s + 0.0157·26-s − 0.192·27-s + 0.719·28-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 | \( 1 + 26.3T + 2.04e3T^{2} \) |
| 5 | \( 1 - 1.24e4T + 4.88e7T^{2} \) |
| 7 | \( 1 + 4.83e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 5.24e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 3.61e4T + 1.79e12T^{2} \) |
| 17 | \( 1 - 2.33e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.72e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 3.09e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 2.08e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 2.18e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 4.80e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 2.86e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.35e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + 2.88e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 2.98e9T + 9.26e18T^{2} \) |
| 61 | \( 1 - 1.17e10T + 4.35e19T^{2} \) |
| 67 | \( 1 + 1.30e10T + 1.22e20T^{2} \) |
| 71 | \( 1 - 4.46e9T + 2.31e20T^{2} \) |
| 73 | \( 1 - 2.16e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 3.67e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 1.84e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 2.23e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 2.89e10T + 7.15e21T^{2} \) |
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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
−10.02536247555824560024915937968, −9.386579295198672086673727107701, −8.543639384224717564193098293820, −6.73908760089506821353388846434, −6.17576202684268865447231922052, −5.11423405510473505241851425073, −3.81773160629511796409720733040, −2.14391643298388916407860183274, −1.15047460592171240084598215357, 0,
1.15047460592171240084598215357, 2.14391643298388916407860183274, 3.81773160629511796409720733040, 5.11423405510473505241851425073, 6.17576202684268865447231922052, 6.73908760089506821353388846434, 8.543639384224717564193098293820, 9.386579295198672086673727107701, 10.02536247555824560024915937968
