| L(s) = 1 | + 2-s + 4-s − 0.404·5-s − 0.595·7-s + 8-s − 0.404·10-s + 5.71·11-s + 2·13-s − 0.595·14-s + 16-s − 3.12·17-s + 5.40·19-s − 0.404·20-s + 5.71·22-s − 8.43·23-s − 4.83·25-s + 2·26-s − 0.595·28-s + 3.40·29-s + 10.3·31-s + 32-s − 3.12·34-s + 0.240·35-s + 7.43·37-s + 5.40·38-s − 0.404·40-s + 8.83·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.180·5-s − 0.225·7-s + 0.353·8-s − 0.127·10-s + 1.72·11-s + 0.554·13-s − 0.159·14-s + 0.250·16-s − 0.756·17-s + 1.23·19-s − 0.0904·20-s + 1.21·22-s − 1.75·23-s − 0.967·25-s + 0.392·26-s − 0.112·28-s + 0.632·29-s + 1.85·31-s + 0.176·32-s − 0.535·34-s + 0.0407·35-s + 1.22·37-s + 0.876·38-s − 0.0639·40-s + 1.38·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 954 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 954 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.590202906\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.590202906\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 53 | \( 1 + T \) |
| good | 5 | \( 1 + 0.404T + 5T^{2} \) |
| 7 | \( 1 + 0.595T + 7T^{2} \) |
| 11 | \( 1 - 5.71T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 3.12T + 17T^{2} \) |
| 19 | \( 1 - 5.40T + 19T^{2} \) |
| 23 | \( 1 + 8.43T + 23T^{2} \) |
| 29 | \( 1 - 3.40T + 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 - 7.43T + 37T^{2} \) |
| 41 | \( 1 - 8.83T + 41T^{2} \) |
| 43 | \( 1 - 7.83T + 43T^{2} \) |
| 47 | \( 1 - 0.808T + 47T^{2} \) |
| 59 | \( 1 + 8.31T + 59T^{2} \) |
| 61 | \( 1 - 4.83T + 61T^{2} \) |
| 67 | \( 1 + 6.83T + 67T^{2} \) |
| 71 | \( 1 + 15.6T + 71T^{2} \) |
| 73 | \( 1 + 3.19T + 73T^{2} \) |
| 79 | \( 1 - 11.9T + 79T^{2} \) |
| 83 | \( 1 + 16.6T + 83T^{2} \) |
| 89 | \( 1 + 8.31T + 89T^{2} \) |
| 97 | \( 1 + 17.8T + 97T^{2} \) |
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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
−9.957146353939088216911900745583, −9.337622271939200750472752441428, −8.273585695822219666724985549064, −7.39969654952741671845026926671, −6.26251303185052360783301902403, −6.01764626575230731355253657466, −4.40913388660116636715398084967, −3.97480109760812859004588911056, −2.75775552531366017523571483998, −1.30059181614807929241065503450,
1.30059181614807929241065503450, 2.75775552531366017523571483998, 3.97480109760812859004588911056, 4.40913388660116636715398084967, 6.01764626575230731355253657466, 6.26251303185052360783301902403, 7.39969654952741671845026926671, 8.273585695822219666724985549064, 9.337622271939200750472752441428, 9.957146353939088216911900745583
