| L(s) = 1 | − 1.61·2-s + 0.618·4-s + 3.85·7-s + 2.23·8-s − 0.618·11-s − 5.47·13-s − 6.23·14-s − 4.85·16-s + 1.47·17-s − 0.854·19-s + 1.00·22-s + 1.85·23-s + 8.85·26-s + 2.38·28-s + 2.76·29-s + 2·31-s + 3.38·32-s − 2.38·34-s + 3·37-s + 1.38·38-s − 6.47·41-s − 0.472·43-s − 0.381·44-s − 3·46-s − 11.6·47-s + 7.85·49-s − 3.38·52-s + ⋯ |
| L(s) = 1 | − 1.14·2-s + 0.309·4-s + 1.45·7-s + 0.790·8-s − 0.186·11-s − 1.51·13-s − 1.66·14-s − 1.21·16-s + 0.357·17-s − 0.195·19-s + 0.213·22-s + 0.386·23-s + 1.73·26-s + 0.450·28-s + 0.513·29-s + 0.359·31-s + 0.597·32-s − 0.408·34-s + 0.493·37-s + 0.224·38-s − 1.01·41-s − 0.0720·43-s − 0.0575·44-s − 0.442·46-s − 1.69·47-s + 1.12·49-s − 0.468·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 7 | \( 1 - 3.85T + 7T^{2} \) |
| 11 | \( 1 + 0.618T + 11T^{2} \) |
| 13 | \( 1 + 5.47T + 13T^{2} \) |
| 17 | \( 1 - 1.47T + 17T^{2} \) |
| 19 | \( 1 + 0.854T + 19T^{2} \) |
| 23 | \( 1 - 1.85T + 23T^{2} \) |
| 29 | \( 1 - 2.76T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 3T + 37T^{2} \) |
| 41 | \( 1 + 6.47T + 41T^{2} \) |
| 43 | \( 1 + 0.472T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 + 8.47T + 53T^{2} \) |
| 59 | \( 1 - 13.9T + 59T^{2} \) |
| 61 | \( 1 + 8.85T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 - 3.32T + 71T^{2} \) |
| 73 | \( 1 + 0.145T + 73T^{2} \) |
| 79 | \( 1 + 6.70T + 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 + 3.61T + 89T^{2} \) |
| 97 | \( 1 + 10.6T + 97T^{2} \) |
| 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
−7.997198260117927697520064820913, −7.35663880831290157186157207127, −6.71822603818540935394082891675, −5.46748248745067043543117992597, −4.78824999450365271891464673845, −4.41030487239414827544981280135, −2.96943269033835306802139366571, −1.99681700103620562341558948571, −1.26235373360684379362149506591, 0,
1.26235373360684379362149506591, 1.99681700103620562341558948571, 2.96943269033835306802139366571, 4.41030487239414827544981280135, 4.78824999450365271891464673845, 5.46748248745067043543117992597, 6.71822603818540935394082891675, 7.35663880831290157186157207127, 7.997198260117927697520064820913
