L(s) = 1 | + 32·2-s + 243·3-s + 1.02e3·4-s + 3.12e3·5-s + 7.77e3·6-s + 1.68e4·7-s + 3.27e4·8-s + 5.90e4·9-s + 1.00e5·10-s + 6.98e5·11-s + 2.48e5·12-s + 2.29e5·13-s + 5.37e5·14-s + 7.59e5·15-s + 1.04e6·16-s + 9.77e6·17-s + 1.88e6·18-s + 1.20e7·19-s + 3.20e6·20-s + 4.08e6·21-s + 2.23e7·22-s − 5.57e7·23-s + 7.96e6·24-s + 9.76e6·25-s + 7.33e6·26-s + 1.43e7·27-s + 1.72e7·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s + 1.30·11-s + 0.288·12-s + 0.171·13-s + 0.267·14-s + 0.258·15-s + 0.250·16-s + 1.66·17-s + 0.235·18-s + 1.11·19-s + 0.223·20-s + 0.218·21-s + 0.924·22-s − 1.80·23-s + 0.204·24-s + 0.199·25-s + 0.121·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(6.819032083\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.819032083\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 32T \) |
| 3 | \( 1 - 243T \) |
| 5 | \( 1 - 3.12e3T \) |
| 7 | \( 1 - 1.68e4T \) |
good | 11 | \( 1 - 6.98e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 2.29e5T + 1.79e12T^{2} \) |
| 17 | \( 1 - 9.77e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 1.20e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 5.57e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.37e8T + 1.22e16T^{2} \) |
| 31 | \( 1 - 2.54e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 3.03e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 1.39e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.85e9T + 9.29e17T^{2} \) |
| 47 | \( 1 - 2.84e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 1.70e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 3.42e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 1.08e10T + 4.35e19T^{2} \) |
| 67 | \( 1 - 2.90e8T + 1.22e20T^{2} \) |
| 71 | \( 1 - 3.15e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + 2.16e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 4.72e9T + 7.47e20T^{2} \) |
| 83 | \( 1 - 8.20e9T + 1.28e21T^{2} \) |
| 89 | \( 1 - 7.16e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 1.14e10T + 7.15e21T^{2} \) |
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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
−10.23761715670179917964592728929, −9.552068483188052170060468292267, −8.315998225678485834964672685107, −7.37859839777016282075531727108, −6.22044722763440522778596529484, −5.30568889524918928707048148403, −4.01101456240679535039804698886, −3.24520096290250700960999465739, −1.89770109376302234754972884116, −1.10630427223516749449499218581,
1.10630427223516749449499218581, 1.89770109376302234754972884116, 3.24520096290250700960999465739, 4.01101456240679535039804698886, 5.30568889524918928707048148403, 6.22044722763440522778596529484, 7.37859839777016282075531727108, 8.315998225678485834964672685107, 9.552068483188052170060468292267, 10.23761715670179917964592728929