| L(s) = 1 | − 63.4·2-s + 243·3-s + 1.98e3·4-s − 1.37e3·5-s − 1.54e4·6-s − 1.61e4·7-s + 4.19e3·8-s + 5.90e4·9-s + 8.72e4·10-s + 3.91e5·11-s + 4.81e5·12-s − 1.24e6·13-s + 1.02e6·14-s − 3.33e5·15-s − 4.32e6·16-s + 8.24e5·17-s − 3.74e6·18-s + 4.80e6·19-s − 2.72e6·20-s − 3.91e6·21-s − 2.48e7·22-s + 2.79e7·23-s + 1.01e6·24-s − 4.69e7·25-s + 7.88e7·26-s + 1.43e7·27-s − 3.19e7·28-s + ⋯ |
| L(s) = 1 | − 1.40·2-s + 0.577·3-s + 0.967·4-s − 0.196·5-s − 0.809·6-s − 0.362·7-s + 0.0452·8-s + 0.333·9-s + 0.275·10-s + 0.733·11-s + 0.558·12-s − 0.928·13-s + 0.508·14-s − 0.113·15-s − 1.03·16-s + 0.140·17-s − 0.467·18-s + 0.444·19-s − 0.190·20-s − 0.209·21-s − 1.02·22-s + 0.905·23-s + 0.0261·24-s − 0.961·25-s + 1.30·26-s + 0.192·27-s − 0.350·28-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 | \( 1 + 63.4T + 2.04e3T^{2} \) |
| 5 | \( 1 + 1.37e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + 1.61e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 3.91e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 1.24e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 8.24e5T + 3.42e13T^{2} \) |
| 19 | \( 1 - 4.80e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 2.79e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 4.28e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + 1.46e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 6.91e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 1.20e9T + 5.50e17T^{2} \) |
| 43 | \( 1 - 3.82e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 1.39e7T + 2.47e18T^{2} \) |
| 53 | \( 1 + 4.64e9T + 9.26e18T^{2} \) |
| 61 | \( 1 + 4.97e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.85e10T + 1.22e20T^{2} \) |
| 71 | \( 1 - 4.49e9T + 2.31e20T^{2} \) |
| 73 | \( 1 - 2.44e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 1.48e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 5.49e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 4.35e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 1.25e11T + 7.15e21T^{2} \) |
| 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
−9.610589458472818937517459641461, −9.436025212924211326034899760749, −8.222148266764613421070165542288, −7.47109154949468466170413839213, −6.54232579890647783852072073266, −4.80736835800237871170973400621, −3.46520285443221386886267387572, −2.19538910436718117894083529947, −1.10941315624682465188586038376, 0,
1.10941315624682465188586038376, 2.19538910436718117894083529947, 3.46520285443221386886267387572, 4.80736835800237871170973400621, 6.54232579890647783852072073266, 7.47109154949468466170413839213, 8.222148266764613421070165542288, 9.436025212924211326034899760749, 9.610589458472818937517459641461
