| L(s) = 1 | + 2-s − 3-s + 4-s + 3.59·5-s − 6-s + 3.34·7-s + 8-s + 9-s + 3.59·10-s − 3.90·11-s − 12-s + 4.00·13-s + 3.34·14-s − 3.59·15-s + 16-s − 17-s + 18-s − 3.67·19-s + 3.59·20-s − 3.34·21-s − 3.90·22-s − 5.58·23-s − 24-s + 7.91·25-s + 4.00·26-s − 27-s + 3.34·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.60·5-s − 0.408·6-s + 1.26·7-s + 0.353·8-s + 0.333·9-s + 1.13·10-s − 1.17·11-s − 0.288·12-s + 1.11·13-s + 0.894·14-s − 0.927·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s − 0.842·19-s + 0.803·20-s − 0.730·21-s − 0.831·22-s − 1.16·23-s − 0.204·24-s + 1.58·25-s + 0.785·26-s − 0.192·27-s + 0.632·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.289201232\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.289201232\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 5 | \( 1 - 3.59T + 5T^{2} \) |
| 7 | \( 1 - 3.34T + 7T^{2} \) |
| 11 | \( 1 + 3.90T + 11T^{2} \) |
| 13 | \( 1 - 4.00T + 13T^{2} \) |
| 19 | \( 1 + 3.67T + 19T^{2} \) |
| 23 | \( 1 + 5.58T + 23T^{2} \) |
| 29 | \( 1 - 2.22T + 29T^{2} \) |
| 31 | \( 1 - 0.830T + 31T^{2} \) |
| 37 | \( 1 - 4.39T + 37T^{2} \) |
| 41 | \( 1 + 0.806T + 41T^{2} \) |
| 43 | \( 1 - 12.0T + 43T^{2} \) |
| 47 | \( 1 + 4.08T + 47T^{2} \) |
| 53 | \( 1 - 8.74T + 53T^{2} \) |
| 61 | \( 1 - 7.47T + 61T^{2} \) |
| 67 | \( 1 - 7.50T + 67T^{2} \) |
| 71 | \( 1 - 3.47T + 71T^{2} \) |
| 73 | \( 1 + 0.169T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 + 16.4T + 83T^{2} \) |
| 89 | \( 1 - 12.1T + 89T^{2} \) |
| 97 | \( 1 - 17.7T + 97T^{2} \) |
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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
−8.106634770124021494501072036069, −7.20607993508146383742577167847, −6.19650524936615930643264203718, −5.97193793528522864846464542875, −5.25010118304835749213637895066, −4.69435569010225410958545191112, −3.86804940907645457330452590843, −2.45455601266496512566813941194, −2.06024302936800896301402439425, −1.07084154415259211426971947153,
1.07084154415259211426971947153, 2.06024302936800896301402439425, 2.45455601266496512566813941194, 3.86804940907645457330452590843, 4.69435569010225410958545191112, 5.25010118304835749213637895066, 5.97193793528522864846464542875, 6.19650524936615930643264203718, 7.20607993508146383742577167847, 8.106634770124021494501072036069
