| L(s) = 1 | − 32·2-s + 566.·3-s + 1.02e3·4-s + 8.35e3·5-s − 1.81e4·6-s − 1.68e4·7-s − 3.27e4·8-s + 1.43e5·9-s − 2.67e5·10-s − 1.55e5·11-s + 5.79e5·12-s + 2.49e6·13-s + 5.37e5·14-s + 4.73e6·15-s + 1.04e6·16-s + 6.27e6·17-s − 4.59e6·18-s − 1.54e7·19-s + 8.55e6·20-s − 9.51e6·21-s + 4.98e6·22-s + 3.08e7·23-s − 1.85e7·24-s + 2.09e7·25-s − 7.99e7·26-s − 1.89e7·27-s − 1.72e7·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.34·3-s + 0.5·4-s + 1.19·5-s − 0.951·6-s − 0.377·7-s − 0.353·8-s + 0.810·9-s − 0.845·10-s − 0.291·11-s + 0.672·12-s + 1.86·13-s + 0.267·14-s + 1.60·15-s + 0.250·16-s + 1.07·17-s − 0.573·18-s − 1.43·19-s + 0.597·20-s − 0.508·21-s + 0.206·22-s + 1.00·23-s − 0.475·24-s + 0.428·25-s − 1.31·26-s − 0.254·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(2.335790794\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.335790794\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 32T \) |
| 7 | \( 1 + 1.68e4T \) |
| good | 3 | \( 1 - 566.T + 1.77e5T^{2} \) |
| 5 | \( 1 - 8.35e3T + 4.88e7T^{2} \) |
| 11 | \( 1 + 1.55e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 2.49e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 6.27e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.54e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 3.08e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 5.84e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 7.55e7T + 2.54e16T^{2} \) |
| 37 | \( 1 + 1.85e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 9.65e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.23e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + 2.09e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 3.38e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 9.92e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 5.12e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 4.09e8T + 1.22e20T^{2} \) |
| 71 | \( 1 + 2.18e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 3.38e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 4.03e9T + 7.47e20T^{2} \) |
| 83 | \( 1 - 9.58e8T + 1.28e21T^{2} \) |
| 89 | \( 1 + 1.95e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 6.10e10T + 7.15e21T^{2} \) |
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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
−17.00301989511529014145403407710, −15.48828125239222771884036663082, −14.07144498877926666892374850554, −13.06050399679577594827157896246, −10.55875754176967958122519628749, −9.274636666285060927185707812084, −8.269344123105698166861756393735, −6.23916855395972690291373589252, −3.12647003899967048568200066316, −1.59310400912638094749936151522,
1.59310400912638094749936151522, 3.12647003899967048568200066316, 6.23916855395972690291373589252, 8.269344123105698166861756393735, 9.274636666285060927185707812084, 10.55875754176967958122519628749, 13.06050399679577594827157896246, 14.07144498877926666892374850554, 15.48828125239222771884036663082, 17.00301989511529014145403407710
