| L(s) = 1 | − 8·5-s + 7·7-s + 56·11-s − 28·13-s + 90·17-s − 74·19-s − 96·23-s − 61·25-s + 222·29-s + 100·31-s − 56·35-s + 58·37-s − 422·41-s − 512·43-s + 148·47-s + 49·49-s + 642·53-s − 448·55-s − 318·59-s + 720·61-s + 224·65-s + 412·67-s + 448·71-s + 994·73-s + 392·77-s + 296·79-s + 386·83-s + ⋯ |
| L(s) = 1 | − 0.715·5-s + 0.377·7-s + 1.53·11-s − 0.597·13-s + 1.28·17-s − 0.893·19-s − 0.870·23-s − 0.487·25-s + 1.42·29-s + 0.579·31-s − 0.270·35-s + 0.257·37-s − 1.60·41-s − 1.81·43-s + 0.459·47-s + 1/7·49-s + 1.66·53-s − 1.09·55-s − 0.701·59-s + 1.51·61-s + 0.427·65-s + 0.751·67-s + 0.748·71-s + 1.59·73-s + 0.580·77-s + 0.421·79-s + 0.510·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.937229344\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.937229344\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 5 | \( 1 + 8 T + p^{3} T^{2} \) |
| 11 | \( 1 - 56 T + p^{3} T^{2} \) |
| 13 | \( 1 + 28 T + p^{3} T^{2} \) |
| 17 | \( 1 - 90 T + p^{3} T^{2} \) |
| 19 | \( 1 + 74 T + p^{3} T^{2} \) |
| 23 | \( 1 + 96 T + p^{3} T^{2} \) |
| 29 | \( 1 - 222 T + p^{3} T^{2} \) |
| 31 | \( 1 - 100 T + p^{3} T^{2} \) |
| 37 | \( 1 - 58 T + p^{3} T^{2} \) |
| 41 | \( 1 + 422 T + p^{3} T^{2} \) |
| 43 | \( 1 + 512 T + p^{3} T^{2} \) |
| 47 | \( 1 - 148 T + p^{3} T^{2} \) |
| 53 | \( 1 - 642 T + p^{3} T^{2} \) |
| 59 | \( 1 + 318 T + p^{3} T^{2} \) |
| 61 | \( 1 - 720 T + p^{3} T^{2} \) |
| 67 | \( 1 - 412 T + p^{3} T^{2} \) |
| 71 | \( 1 - 448 T + p^{3} T^{2} \) |
| 73 | \( 1 - 994 T + p^{3} T^{2} \) |
| 79 | \( 1 - 296 T + p^{3} T^{2} \) |
| 83 | \( 1 - 386 T + p^{3} T^{2} \) |
| 89 | \( 1 - 6 T + p^{3} T^{2} \) |
| 97 | \( 1 + 138 T + p^{3} T^{2} \) |
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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
−9.726222206168934119996004370124, −8.500241272846391722349115977093, −8.111792430184517981207288348631, −7.01608524368968004731295568174, −6.32721277479922267292279288987, −5.13765587072913599981327523837, −4.17499101618705271414933126438, −3.45534619778237441812378159625, −1.99163381467494039665983200855, −0.75671235558084490522839579786,
0.75671235558084490522839579786, 1.99163381467494039665983200855, 3.45534619778237441812378159625, 4.17499101618705271414933126438, 5.13765587072913599981327523837, 6.32721277479922267292279288987, 7.01608524368968004731295568174, 8.111792430184517981207288348631, 8.500241272846391722349115977093, 9.726222206168934119996004370124
