| L(s) = 1 | − 1.41·5-s + 2·7-s + 1.41·11-s + 1.41·13-s − 2·17-s + 4.24·19-s − 6·23-s − 2.99·25-s − 4.24·29-s + 8·31-s − 2.82·35-s − 4.24·37-s − 7.07·43-s − 8·47-s − 3·49-s + 7.07·53-s − 2.00·55-s + 4.24·59-s − 12.7·61-s − 2.00·65-s + 7.07·67-s − 10·71-s + 4·73-s + 2.82·77-s + 1.41·83-s + 2.82·85-s − 4·89-s + ⋯ |
| L(s) = 1 | − 0.632·5-s + 0.755·7-s + 0.426·11-s + 0.392·13-s − 0.485·17-s + 0.973·19-s − 1.25·23-s − 0.599·25-s − 0.787·29-s + 1.43·31-s − 0.478·35-s − 0.697·37-s − 1.07·43-s − 1.16·47-s − 0.428·49-s + 0.971·53-s − 0.269·55-s + 0.552·59-s − 1.62·61-s − 0.248·65-s + 0.863·67-s − 1.18·71-s + 0.468·73-s + 0.322·77-s + 0.155·83-s + 0.306·85-s − 0.423·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 1.41T + 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 - 1.41T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 4.24T + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 4.24T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 4.24T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 7.07T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 7.07T + 53T^{2} \) |
| 59 | \( 1 - 4.24T + 59T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 - 7.07T + 67T^{2} \) |
| 71 | \( 1 + 10T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 1.41T + 83T^{2} \) |
| 89 | \( 1 + 4T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
| 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
−7.48654113349428805147724410845, −6.73082787597525248820353617691, −6.03865837456829704952567991427, −5.24904918684148510310328686897, −4.52351514702833609690323614937, −3.87882403593986685800817551240, −3.19423179007541018340507254649, −2.05668401343670540566030108067, −1.29403526790407359603601570518, 0,
1.29403526790407359603601570518, 2.05668401343670540566030108067, 3.19423179007541018340507254649, 3.87882403593986685800817551240, 4.52351514702833609690323614937, 5.24904918684148510310328686897, 6.03865837456829704952567991427, 6.73082787597525248820353617691, 7.48654113349428805147724410845
