| L(s) = 1 | − 2-s + 2.50·3-s + 4-s + 0.355·5-s − 2.50·6-s − 0.987·7-s − 8-s + 3.27·9-s − 0.355·10-s + 0.109·11-s + 2.50·12-s + 6.22·13-s + 0.987·14-s + 0.889·15-s + 16-s + 4.42·17-s − 3.27·18-s − 1.87·19-s + 0.355·20-s − 2.47·21-s − 0.109·22-s + 4.83·23-s − 2.50·24-s − 4.87·25-s − 6.22·26-s + 0.683·27-s − 0.987·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.44·3-s + 0.5·4-s + 0.158·5-s − 1.02·6-s − 0.373·7-s − 0.353·8-s + 1.09·9-s − 0.112·10-s + 0.0329·11-s + 0.723·12-s + 1.72·13-s + 0.263·14-s + 0.229·15-s + 0.250·16-s + 1.07·17-s − 0.771·18-s − 0.429·19-s + 0.0794·20-s − 0.539·21-s − 0.0233·22-s + 1.00·23-s − 0.511·24-s − 0.974·25-s − 1.22·26-s + 0.131·27-s − 0.186·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.829906012\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.829906012\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 - 2.50T + 3T^{2} \) |
| 5 | \( 1 - 0.355T + 5T^{2} \) |
| 7 | \( 1 + 0.987T + 7T^{2} \) |
| 11 | \( 1 - 0.109T + 11T^{2} \) |
| 13 | \( 1 - 6.22T + 13T^{2} \) |
| 17 | \( 1 - 4.42T + 17T^{2} \) |
| 19 | \( 1 + 1.87T + 19T^{2} \) |
| 23 | \( 1 - 4.83T + 23T^{2} \) |
| 29 | \( 1 - 0.947T + 29T^{2} \) |
| 37 | \( 1 - 4.10T + 37T^{2} \) |
| 41 | \( 1 + 5.26T + 41T^{2} \) |
| 43 | \( 1 - 9.41T + 43T^{2} \) |
| 47 | \( 1 + 0.558T + 47T^{2} \) |
| 53 | \( 1 + 0.791T + 53T^{2} \) |
| 59 | \( 1 + 6.78T + 59T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 - 0.420T + 67T^{2} \) |
| 71 | \( 1 - 16.4T + 71T^{2} \) |
| 73 | \( 1 - 0.0523T + 73T^{2} \) |
| 79 | \( 1 - 2.91T + 79T^{2} \) |
| 83 | \( 1 - 1.72T + 83T^{2} \) |
| 89 | \( 1 - 11.5T + 89T^{2} \) |
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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
−8.120142559328396213817830948219, −7.76450630836447627658926489920, −6.80123956986971981499738371963, −6.17504002128488858806042504225, −5.34959497044053121183374488995, −3.98423101510662996104220839958, −3.48988385242157306335215401626, −2.76667473431990354401109279761, −1.85199199282964927726859808256, −0.964205706536893273306540135566,
0.964205706536893273306540135566, 1.85199199282964927726859808256, 2.76667473431990354401109279761, 3.48988385242157306335215401626, 3.98423101510662996104220839958, 5.34959497044053121183374488995, 6.17504002128488858806042504225, 6.80123956986971981499738371963, 7.76450630836447627658926489920, 8.120142559328396213817830948219
