| L(s) = 1 | − 58.2·2-s − 243·3-s + 1.34e3·4-s + 1.17e4·5-s + 1.41e4·6-s + 2.45e4·7-s + 4.10e4·8-s + 5.90e4·9-s − 6.85e5·10-s − 9.38e5·11-s − 3.26e5·12-s − 1.32e6·13-s − 1.43e6·14-s − 2.85e6·15-s − 5.14e6·16-s + 5.00e5·17-s − 3.43e6·18-s − 1.82e7·19-s + 1.57e7·20-s − 5.97e6·21-s + 5.46e7·22-s − 3.11e6·23-s − 9.98e6·24-s + 8.95e7·25-s + 7.74e7·26-s − 1.43e7·27-s + 3.30e7·28-s + ⋯ |
| L(s) = 1 | − 1.28·2-s − 0.577·3-s + 0.655·4-s + 1.68·5-s + 0.742·6-s + 0.552·7-s + 0.443·8-s + 0.333·9-s − 2.16·10-s − 1.75·11-s − 0.378·12-s − 0.992·13-s − 0.711·14-s − 0.972·15-s − 1.22·16-s + 0.0855·17-s − 0.428·18-s − 1.69·19-s + 1.10·20-s − 0.319·21-s + 2.26·22-s − 0.100·23-s − 0.255·24-s + 1.83·25-s + 1.27·26-s − 0.192·27-s + 0.362·28-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
| \(L(6)\) |
\(\approx\) |
\(0.6785171520\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6785171520\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 + 58.2T + 2.04e3T^{2} \) |
| 5 | \( 1 - 1.17e4T + 4.88e7T^{2} \) |
| 7 | \( 1 - 2.45e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 9.38e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 1.32e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 5.00e5T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.82e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 3.11e6T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.34e8T + 1.22e16T^{2} \) |
| 31 | \( 1 - 1.46e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 7.84e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 8.56e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.12e9T + 9.29e17T^{2} \) |
| 47 | \( 1 - 1.67e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 1.57e9T + 9.26e18T^{2} \) |
| 61 | \( 1 - 1.01e10T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.64e10T + 1.22e20T^{2} \) |
| 71 | \( 1 - 1.39e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 2.39e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 4.32e9T + 7.47e20T^{2} \) |
| 83 | \( 1 - 4.65e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 3.36e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 1.01e11T + 7.15e21T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.32274122604648584858033612961, −9.911348710452175737182858685744, −8.785279299149834227674840857203, −7.81224690337722131491483978522, −6.71532046148091186745599482911, −5.48755895445109827160918241248, −4.78674360332369612827780535733, −2.32136169588672391161764108844, −1.85342878344987950976812155268, −0.45336343576841676752129191905,
0.45336343576841676752129191905, 1.85342878344987950976812155268, 2.32136169588672391161764108844, 4.78674360332369612827780535733, 5.48755895445109827160918241248, 6.71532046148091186745599482911, 7.81224690337722131491483978522, 8.785279299149834227674840857203, 9.911348710452175737182858685744, 10.32274122604648584858033612961
