| L(s) = 1 | − 0.915·3-s − 1.31·5-s − 2.16·9-s + 2.72·11-s + 6.20·13-s + 1.19·15-s − 6.76·17-s + 5.15·19-s + 6.76·23-s − 3.28·25-s + 4.72·27-s − 6.37·29-s + 2.28·31-s − 2.49·33-s + 9.35·37-s − 5.67·39-s − 41-s − 6.74·43-s + 2.83·45-s + 8.15·47-s + 6.19·51-s − 12.5·53-s − 3.57·55-s − 4.71·57-s + 1.48·59-s − 2.36·61-s − 8.12·65-s + ⋯ |
| L(s) = 1 | − 0.528·3-s − 0.585·5-s − 0.720·9-s + 0.822·11-s + 1.72·13-s + 0.309·15-s − 1.64·17-s + 1.18·19-s + 1.40·23-s − 0.656·25-s + 0.909·27-s − 1.18·29-s + 0.411·31-s − 0.434·33-s + 1.53·37-s − 0.909·39-s − 0.156·41-s − 1.02·43-s + 0.422·45-s + 1.18·47-s + 0.867·51-s − 1.72·53-s − 0.481·55-s − 0.625·57-s + 0.193·59-s − 0.303·61-s − 1.00·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.390810505\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.390810505\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 + 0.915T + 3T^{2} \) |
| 5 | \( 1 + 1.31T + 5T^{2} \) |
| 11 | \( 1 - 2.72T + 11T^{2} \) |
| 13 | \( 1 - 6.20T + 13T^{2} \) |
| 17 | \( 1 + 6.76T + 17T^{2} \) |
| 19 | \( 1 - 5.15T + 19T^{2} \) |
| 23 | \( 1 - 6.76T + 23T^{2} \) |
| 29 | \( 1 + 6.37T + 29T^{2} \) |
| 31 | \( 1 - 2.28T + 31T^{2} \) |
| 37 | \( 1 - 9.35T + 37T^{2} \) |
| 43 | \( 1 + 6.74T + 43T^{2} \) |
| 47 | \( 1 - 8.15T + 47T^{2} \) |
| 53 | \( 1 + 12.5T + 53T^{2} \) |
| 59 | \( 1 - 1.48T + 59T^{2} \) |
| 61 | \( 1 + 2.36T + 61T^{2} \) |
| 67 | \( 1 - 2.61T + 67T^{2} \) |
| 71 | \( 1 + 1.28T + 71T^{2} \) |
| 73 | \( 1 - 3.81T + 73T^{2} \) |
| 79 | \( 1 - 4.28T + 79T^{2} \) |
| 83 | \( 1 + 2.35T + 83T^{2} \) |
| 89 | \( 1 - 3.98T + 89T^{2} \) |
| 97 | \( 1 + 7.03T + 97T^{2} \) |
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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
−7.88720591773302043400922043744, −6.99892499210005384210578190757, −6.39945152690651324169274816995, −5.87239319913383237689784552005, −5.06238438586256686012939792512, −4.23235677133748672325932928624, −3.59498431246669826641392155495, −2.82629446824335236757596187743, −1.55861840585172288153663749257, −0.62952167572649113325085159030,
0.62952167572649113325085159030, 1.55861840585172288153663749257, 2.82629446824335236757596187743, 3.59498431246669826641392155495, 4.23235677133748672325932928624, 5.06238438586256686012939792512, 5.87239319913383237689784552005, 6.39945152690651324169274816995, 6.99892499210005384210578190757, 7.88720591773302043400922043744
