| L(s) = 1 | + (−3 + 3i)5-s − 3i·9-s + (−1 − i)13-s − 8·17-s − 13i·25-s + (−7 − 7i)29-s + (−5 + 5i)37-s + 8i·41-s + (9 + 9i)45-s + 7·49-s + (−5 + 5i)53-s + (−1 − i)61-s + 6·65-s + 6i·73-s − 9·81-s + ⋯ |
| L(s) = 1 | + (−1.34 + 1.34i)5-s − i·9-s + (−0.277 − 0.277i)13-s − 1.94·17-s − 2.60i·25-s + (−1.29 − 1.29i)29-s + (−0.821 + 0.821i)37-s + 1.24i·41-s + (1.34 + 1.34i)45-s + 49-s + (−0.686 + 0.686i)53-s + (−0.128 − 0.128i)61-s + 0.744·65-s + 0.702i·73-s − 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 + 3iT^{2} \) |
| 5 | \( 1 + (3 - 3i)T - 5iT^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 11iT^{2} \) |
| 13 | \( 1 + (1 + i)T + 13iT^{2} \) |
| 17 | \( 1 + 8T + 17T^{2} \) |
| 19 | \( 1 + 19iT^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + (7 + 7i)T + 29iT^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + (5 - 5i)T - 37iT^{2} \) |
| 41 | \( 1 - 8iT - 41T^{2} \) |
| 43 | \( 1 - 43iT^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + (5 - 5i)T - 53iT^{2} \) |
| 59 | \( 1 - 59iT^{2} \) |
| 61 | \( 1 + (1 + i)T + 61iT^{2} \) |
| 67 | \( 1 + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83iT^{2} \) |
| 89 | \( 1 - 10iT - 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
| 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
−10.72756617107388021907404876625, −9.694846797595585628577505532914, −8.632234507736590101961628433772, −7.66700851176729134844715704013, −6.86681513196180977535606462490, −6.20005292841791693694369192528, −4.44215588362433810879658579021, −3.63749608110412046786385718474, −2.57789197994211068748399544012, 0,
1.94235188945846926758769502317, 3.78391857123772335592102552475, 4.64151812143918439234263580506, 5.35429359836309575592434763750, 7.03123875703169337427194047700, 7.71844317111024991511995539038, 8.763736629208434220392247529823, 9.069540656469781468452912457701, 10.68086206437723389100626391399
