L(s) = 1 | + 41.1·2-s − 243·3-s − 354.·4-s − 4.33e3·5-s − 9.99e3·6-s + 6.42e4·7-s − 9.88e4·8-s + 5.90e4·9-s − 1.78e5·10-s − 5.41e5·11-s + 8.62e4·12-s + 2.89e5·13-s + 2.64e6·14-s + 1.05e6·15-s − 3.34e6·16-s − 5.10e6·17-s + 2.42e6·18-s + 7.74e6·19-s + 1.53e6·20-s − 1.56e7·21-s − 2.22e7·22-s − 1.12e7·23-s + 2.40e7·24-s − 3.00e7·25-s + 1.19e7·26-s − 1.43e7·27-s − 2.28e7·28-s + ⋯ |
L(s) = 1 | + 0.909·2-s − 0.577·3-s − 0.173·4-s − 0.620·5-s − 0.524·6-s + 1.44·7-s − 1.06·8-s + 0.333·9-s − 0.564·10-s − 1.01·11-s + 0.100·12-s + 0.216·13-s + 1.31·14-s + 0.358·15-s − 0.796·16-s − 0.872·17-s + 0.303·18-s + 0.717·19-s + 0.107·20-s − 0.834·21-s − 0.921·22-s − 0.365·23-s + 0.615·24-s − 0.614·25-s + 0.196·26-s − 0.192·27-s − 0.250·28-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
\(L(6)\) |
\(\approx\) |
\(1.653582898\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.653582898\) |
\(L(\frac{13}{2})\) |
|
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 | \( 1 - 41.1T + 2.04e3T^{2} \) |
| 5 | \( 1 + 4.33e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 6.42e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 5.41e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 2.89e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + 5.10e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 7.74e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 1.12e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.00e8T + 1.22e16T^{2} \) |
| 31 | \( 1 - 1.75e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 4.23e7T + 1.77e17T^{2} \) |
| 41 | \( 1 - 9.20e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.57e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + 3.05e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 2.66e9T + 9.26e18T^{2} \) |
| 61 | \( 1 + 5.17e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.91e10T + 1.22e20T^{2} \) |
| 71 | \( 1 - 1.17e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 1.77e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 3.41e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 1.13e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 5.07e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 1.37e11T + 7.15e21T^{2} \) |
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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
−11.21541741477397385979589665943, −9.799319715618042651440489970096, −8.389944663754503298387014254817, −7.68676834732667013987848704226, −6.21457822049637522802664988267, −5.07641989581766571913763537319, −4.64359011147755775531996921735, −3.49413030470977198079248293316, −2.01369349868715406851623262458, −0.51420892784777786683489124316,
0.51420892784777786683489124316, 2.01369349868715406851623262458, 3.49413030470977198079248293316, 4.64359011147755775531996921735, 5.07641989581766571913763537319, 6.21457822049637522802664988267, 7.68676834732667013987848704226, 8.389944663754503298387014254817, 9.799319715618042651440489970096, 11.21541741477397385979589665943