| L(s) = 1 | + 2.78·3-s + 0.592·5-s + 4.73·9-s − 2.21·11-s − 5.00·13-s + 1.64·15-s − 4.32·17-s + 2.38·19-s + 1.67·23-s − 4.64·25-s + 4.82·27-s − 1.27·29-s − 2.49·31-s − 6.15·33-s − 4.24·37-s − 13.9·39-s − 41-s − 8.15·43-s + 2.80·45-s − 1.65·47-s − 12.0·51-s − 7.60·53-s − 1.31·55-s + 6.63·57-s + 6.12·59-s − 5.08·61-s − 2.96·65-s + ⋯ |
| L(s) = 1 | + 1.60·3-s + 0.265·5-s + 1.57·9-s − 0.667·11-s − 1.38·13-s + 0.425·15-s − 1.04·17-s + 0.547·19-s + 0.348·23-s − 0.929·25-s + 0.927·27-s − 0.236·29-s − 0.448·31-s − 1.07·33-s − 0.697·37-s − 2.23·39-s − 0.156·41-s − 1.24·43-s + 0.418·45-s − 0.242·47-s − 1.68·51-s − 1.04·53-s − 0.176·55-s + 0.878·57-s + 0.797·59-s − 0.650·61-s − 0.368·65-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 | \( 1 - 2.78T + 3T^{2} \) |
| 5 | \( 1 - 0.592T + 5T^{2} \) |
| 11 | \( 1 + 2.21T + 11T^{2} \) |
| 13 | \( 1 + 5.00T + 13T^{2} \) |
| 17 | \( 1 + 4.32T + 17T^{2} \) |
| 19 | \( 1 - 2.38T + 19T^{2} \) |
| 23 | \( 1 - 1.67T + 23T^{2} \) |
| 29 | \( 1 + 1.27T + 29T^{2} \) |
| 31 | \( 1 + 2.49T + 31T^{2} \) |
| 37 | \( 1 + 4.24T + 37T^{2} \) |
| 43 | \( 1 + 8.15T + 43T^{2} \) |
| 47 | \( 1 + 1.65T + 47T^{2} \) |
| 53 | \( 1 + 7.60T + 53T^{2} \) |
| 59 | \( 1 - 6.12T + 59T^{2} \) |
| 61 | \( 1 + 5.08T + 61T^{2} \) |
| 67 | \( 1 + 14.0T + 67T^{2} \) |
| 71 | \( 1 + 2.62T + 71T^{2} \) |
| 73 | \( 1 - 16.2T + 73T^{2} \) |
| 79 | \( 1 - 4.77T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 + 6.37T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.50440108396023430021274462867, −7.16640120639576708909823310597, −6.21638636992193188967725054702, −5.16419821322606060614603395703, −4.65545845687515627479167429788, −3.67234208480226290576368862286, −3.00860999259595689937906605899, −2.27998151601237220621441626752, −1.74349039067530723182622230281, 0,
1.74349039067530723182622230281, 2.27998151601237220621441626752, 3.00860999259595689937906605899, 3.67234208480226290576368862286, 4.65545845687515627479167429788, 5.16419821322606060614603395703, 6.21638636992193188967725054702, 7.16640120639576708909823310597, 7.50440108396023430021274462867
