| L(s) = 1 | − 6.16·2-s + 7·3-s + 5.99·4-s − 19·5-s − 43.1·6-s + 49.3·7-s + 160.·8-s − 194·9-s + 117.·10-s + 41.9·12-s + 739.·13-s − 303.·14-s − 133·15-s − 1.18e3·16-s − 1.82e3·17-s + 1.19e3·18-s + 2.12e3·19-s − 113.·20-s + 345.·21-s + 3.07e3·23-s + 1.12e3·24-s − 2.76e3·25-s − 4.56e3·26-s − 3.05e3·27-s + 295.·28-s − 493.·29-s + 819.·30-s + ⋯ |
| L(s) = 1 | − 1.08·2-s + 0.449·3-s + 0.187·4-s − 0.339·5-s − 0.489·6-s + 0.380·7-s + 0.885·8-s − 0.798·9-s + 0.370·10-s + 0.0841·12-s + 1.21·13-s − 0.414·14-s − 0.152·15-s − 1.15·16-s − 1.53·17-s + 0.869·18-s + 1.34·19-s − 0.0637·20-s + 0.170·21-s + 1.21·23-s + 0.397·24-s − 0.884·25-s − 1.32·26-s − 0.807·27-s + 0.0713·28-s − 0.108·29-s + 0.166·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| good | 2 | \( 1 + 6.16T + 32T^{2} \) |
| 3 | \( 1 - 7T + 243T^{2} \) |
| 5 | \( 1 + 19T + 3.12e3T^{2} \) |
| 7 | \( 1 - 49.3T + 1.68e4T^{2} \) |
| 13 | \( 1 - 739.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.82e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.12e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.07e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 493.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.58e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.14e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.74e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.41e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.66e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.62e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.45e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 7.79e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.06e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.10e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.36e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.37e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.48e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.09e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.36e4T + 8.58e9T^{2} \) |
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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
−11.51115705876298014244318773416, −10.98561933365643465774322488000, −9.586847145289665824475545738332, −8.674097786432749088081036344780, −8.118781378016770780858446352580, −6.82662712442880588104621251740, −5.06356317718718722950276249484, −3.46437978352597760446699600659, −1.62353433597525133202022977064, 0,
1.62353433597525133202022977064, 3.46437978352597760446699600659, 5.06356317718718722950276249484, 6.82662712442880588104621251740, 8.118781378016770780858446352580, 8.674097786432749088081036344780, 9.586847145289665824475545738332, 10.98561933365643465774322488000, 11.51115705876298014244318773416
