| L(s) = 1 | − 3.52·2-s − 19.6·4-s + 25·5-s − 80.4·7-s + 181.·8-s − 88.0·10-s + 520.·11-s + 421.·13-s + 283.·14-s − 12.3·16-s + 1.16e3·17-s + 1.17e3·19-s − 490.·20-s − 1.83e3·22-s + 2.37e3·23-s + 625·25-s − 1.48e3·26-s + 1.57e3·28-s − 7.05e3·29-s − 5.56e3·31-s − 5.77e3·32-s − 4.09e3·34-s − 2.01e3·35-s + 2.38e3·37-s − 4.12e3·38-s + 4.54e3·40-s + 8.38e3·41-s + ⋯ |
| L(s) = 1 | − 0.622·2-s − 0.612·4-s + 0.447·5-s − 0.620·7-s + 1.00·8-s − 0.278·10-s + 1.29·11-s + 0.692·13-s + 0.386·14-s − 0.0120·16-s + 0.975·17-s + 0.743·19-s − 0.273·20-s − 0.807·22-s + 0.935·23-s + 0.200·25-s − 0.430·26-s + 0.380·28-s − 1.55·29-s − 1.03·31-s − 0.996·32-s − 0.607·34-s − 0.277·35-s + 0.287·37-s − 0.463·38-s + 0.448·40-s + 0.779·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.123897691\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.123897691\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| good | 2 | \( 1 + 3.52T + 32T^{2} \) |
| 7 | \( 1 + 80.4T + 1.68e4T^{2} \) |
| 11 | \( 1 - 520.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 421.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.16e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.17e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.37e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.05e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.56e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.38e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 8.38e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.05e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 8.35e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.83e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.75e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.73e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.58e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.15e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.68e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.00e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.32e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.07e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.24e4T + 8.58e9T^{2} \) |
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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
−14.66784970969022019510655850936, −13.68496200415498815155107445593, −12.61180369256250028057802710003, −11.01034771856315258632575862221, −9.596994509059774957244181837257, −9.023182964180726345828141504177, −7.35453074034000674602867122277, −5.70691639151721770956250193902, −3.76406847311085762487158546967, −1.11486975673290849705631102973,
1.11486975673290849705631102973, 3.76406847311085762487158546967, 5.70691639151721770956250193902, 7.35453074034000674602867122277, 9.023182964180726345828141504177, 9.596994509059774957244181837257, 11.01034771856315258632575862221, 12.61180369256250028057802710003, 13.68496200415498815155107445593, 14.66784970969022019510655850936
