L(s) = 1 | − 0.475·3-s − 1.19·5-s − 2.77·9-s − 4.51·11-s − 1.46·13-s + 0.569·15-s + 4.96·17-s + 3.69·19-s + 4.36·23-s − 3.56·25-s + 2.74·27-s − 3.12·29-s + 7.79·31-s + 2.14·33-s + 9.52·37-s + 0.698·39-s − 41-s − 11.7·43-s + 3.31·45-s − 6.93·47-s − 2.36·51-s + 8.63·53-s + 5.39·55-s − 1.75·57-s + 2.99·59-s − 0.624·61-s + 1.75·65-s + ⋯ |
L(s) = 1 | − 0.274·3-s − 0.534·5-s − 0.924·9-s − 1.36·11-s − 0.407·13-s + 0.146·15-s + 1.20·17-s + 0.846·19-s + 0.909·23-s − 0.713·25-s + 0.528·27-s − 0.580·29-s + 1.39·31-s + 0.373·33-s + 1.56·37-s + 0.111·39-s − 0.156·41-s − 1.78·43-s + 0.494·45-s − 1.01·47-s − 0.331·51-s + 1.18·53-s + 0.728·55-s − 0.232·57-s + 0.389·59-s − 0.0799·61-s + 0.217·65-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.475T + 3T^{2} \) |
| 5 | \( 1 + 1.19T + 5T^{2} \) |
| 11 | \( 1 + 4.51T + 11T^{2} \) |
| 13 | \( 1 + 1.46T + 13T^{2} \) |
| 17 | \( 1 - 4.96T + 17T^{2} \) |
| 19 | \( 1 - 3.69T + 19T^{2} \) |
| 23 | \( 1 - 4.36T + 23T^{2} \) |
| 29 | \( 1 + 3.12T + 29T^{2} \) |
| 31 | \( 1 - 7.79T + 31T^{2} \) |
| 37 | \( 1 - 9.52T + 37T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 + 6.93T + 47T^{2} \) |
| 53 | \( 1 - 8.63T + 53T^{2} \) |
| 59 | \( 1 - 2.99T + 59T^{2} \) |
| 61 | \( 1 + 0.624T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 - 1.26T + 71T^{2} \) |
| 73 | \( 1 - 6.30T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 - 7.11T + 83T^{2} \) |
| 89 | \( 1 - 10.5T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−7.72649890604112082824491873601, −6.83826497099685635540923058701, −5.99972225711004483075096014641, −5.20220914893116216072026681180, −5.02223808619525378764179743943, −3.79500091288163269975730009114, −3.03424719297058183123810829750, −2.47295617542385002919829168519, −1.03009251481172247951656997382, 0,
1.03009251481172247951656997382, 2.47295617542385002919829168519, 3.03424719297058183123810829750, 3.79500091288163269975730009114, 5.02223808619525378764179743943, 5.20220914893116216072026681180, 5.99972225711004483075096014641, 6.83826497099685635540923058701, 7.72649890604112082824491873601