| L(s) = 1 | + 2.49·2-s − 2.82·3-s + 4.21·4-s + 2.51·5-s − 7.03·6-s − 7-s + 5.51·8-s + 4.95·9-s + 6.27·10-s − 5.25·11-s − 11.8·12-s − 4.20·13-s − 2.49·14-s − 7.10·15-s + 5.31·16-s + 17-s + 12.3·18-s + 0.781·19-s + 10.6·20-s + 2.82·21-s − 13.1·22-s + 2.42·23-s − 15.5·24-s + 1.34·25-s − 10.4·26-s − 5.52·27-s − 4.21·28-s + ⋯ |
| L(s) = 1 | + 1.76·2-s − 1.62·3-s + 2.10·4-s + 1.12·5-s − 2.87·6-s − 0.377·7-s + 1.94·8-s + 1.65·9-s + 1.98·10-s − 1.58·11-s − 3.42·12-s − 1.16·13-s − 0.666·14-s − 1.83·15-s + 1.32·16-s + 0.242·17-s + 2.91·18-s + 0.179·19-s + 2.37·20-s + 0.615·21-s − 2.79·22-s + 0.505·23-s − 3.17·24-s + 0.268·25-s − 2.05·26-s − 1.06·27-s − 0.795·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.785346863\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.785346863\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 - 2.49T + 2T^{2} \) |
| 3 | \( 1 + 2.82T + 3T^{2} \) |
| 5 | \( 1 - 2.51T + 5T^{2} \) |
| 11 | \( 1 + 5.25T + 11T^{2} \) |
| 13 | \( 1 + 4.20T + 13T^{2} \) |
| 19 | \( 1 - 0.781T + 19T^{2} \) |
| 23 | \( 1 - 2.42T + 23T^{2} \) |
| 29 | \( 1 - 3.42T + 29T^{2} \) |
| 31 | \( 1 - 6.94T + 31T^{2} \) |
| 37 | \( 1 + 7.02T + 37T^{2} \) |
| 41 | \( 1 - 1.83T + 41T^{2} \) |
| 43 | \( 1 - 8.63T + 43T^{2} \) |
| 47 | \( 1 + 1.34T + 47T^{2} \) |
| 53 | \( 1 + 1.95T + 53T^{2} \) |
| 59 | \( 1 - 2.43T + 59T^{2} \) |
| 61 | \( 1 - 6.16T + 61T^{2} \) |
| 67 | \( 1 - 1.53T + 67T^{2} \) |
| 71 | \( 1 + 1.65T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 - 4.49T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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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
−13.24497822038478251826550352038, −12.60713325520642952974984619053, −11.83408946003397426673002437221, −10.64667234792695529590066683618, −9.998150408717924400876546765722, −7.25051521859539390629815051782, −6.19959789788314820696615373651, −5.40656277677711083732661634593, −4.79444240404426559479454414831, −2.60572208685349216938721988861,
2.60572208685349216938721988861, 4.79444240404426559479454414831, 5.40656277677711083732661634593, 6.19959789788314820696615373651, 7.25051521859539390629815051782, 9.998150408717924400876546765722, 10.64667234792695529590066683618, 11.83408946003397426673002437221, 12.60713325520642952974984619053, 13.24497822038478251826550352038
