L(s) = 1 | − 78.8·2-s + 243·3-s + 4.16e3·4-s + 6.89e3·5-s − 1.91e4·6-s + 4.11e4·7-s − 1.66e5·8-s + 5.90e4·9-s − 5.43e5·10-s − 4.69e5·11-s + 1.01e6·12-s + 1.92e6·13-s − 3.24e6·14-s + 1.67e6·15-s + 4.61e6·16-s − 6.53e6·17-s − 4.65e6·18-s + 1.40e7·19-s + 2.87e7·20-s + 9.99e6·21-s + 3.70e7·22-s + 4.11e6·23-s − 4.05e7·24-s − 1.29e6·25-s − 1.52e8·26-s + 1.43e7·27-s + 1.71e8·28-s + ⋯ |
L(s) = 1 | − 1.74·2-s + 0.577·3-s + 2.03·4-s + 0.986·5-s − 1.00·6-s + 0.924·7-s − 1.79·8-s + 0.333·9-s − 1.71·10-s − 0.879·11-s + 1.17·12-s + 1.44·13-s − 1.61·14-s + 0.569·15-s + 1.10·16-s − 1.11·17-s − 0.580·18-s + 1.30·19-s + 2.00·20-s + 0.534·21-s + 1.53·22-s + 0.133·23-s − 1.03·24-s − 0.0266·25-s − 2.50·26-s + 0.192·27-s + 1.88·28-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 | \( 1 + 78.8T + 2.04e3T^{2} \) |
| 5 | \( 1 - 6.89e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 4.11e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 4.69e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 1.92e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 6.53e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 1.40e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 4.11e6T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.20e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 6.86e7T + 2.54e16T^{2} \) |
| 37 | \( 1 + 4.59e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 9.53e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.75e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + 7.20e8T + 2.47e18T^{2} \) |
| 53 | \( 1 + 4.93e9T + 9.26e18T^{2} \) |
| 61 | \( 1 + 6.74e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 8.57e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 3.87e8T + 2.31e20T^{2} \) |
| 73 | \( 1 + 6.36e9T + 3.13e20T^{2} \) |
| 79 | \( 1 - 9.47e9T + 7.47e20T^{2} \) |
| 83 | \( 1 - 2.58e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 5.07e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 2.09e10T + 7.15e21T^{2} \) |
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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
−9.953936215348964811080395296947, −9.070574176346674680398265782096, −8.376640584506696967431315844119, −7.55469441951552112625036457567, −6.42054506552825795329115919217, −5.13981590624280698162707146451, −3.19203274865512287626602272819, −1.79091505268608845254314627172, −1.56430324487283690159848899121, 0,
1.56430324487283690159848899121, 1.79091505268608845254314627172, 3.19203274865512287626602272819, 5.13981590624280698162707146451, 6.42054506552825795329115919217, 7.55469441951552112625036457567, 8.376640584506696967431315844119, 9.070574176346674680398265782096, 9.953936215348964811080395296947