| L(s) = 1 | − 1.41i·3-s − 2.64·7-s + 0.999·9-s + 3.74i·11-s + 4.24i·13-s − 5.29·19-s + 3.74i·21-s + (3 − 3.74i)23-s − 5·25-s − 5.65i·27-s + 6·29-s − 8.48i·31-s + 5.29·33-s − 11.2i·37-s + 6·39-s + ⋯ |
| L(s) = 1 | − 0.816i·3-s − 0.999·7-s + 0.333·9-s + 1.12i·11-s + 1.17i·13-s − 1.21·19-s + 0.816i·21-s + (0.625 − 0.780i)23-s − 25-s − 1.08i·27-s + 1.11·29-s − 1.52i·31-s + 0.921·33-s − 1.84i·37-s + 0.960·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.625 + 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.625 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9810838261\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9810838261\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 2.64T \) |
| 23 | \( 1 + (-3 + 3.74i)T \) |
| good | 3 | \( 1 + 1.41iT - 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 - 3.74iT - 11T^{2} \) |
| 13 | \( 1 - 4.24iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 5.29T + 19T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 8.48iT - 31T^{2} \) |
| 37 | \( 1 + 11.2iT - 37T^{2} \) |
| 41 | \( 1 + 5.65iT - 41T^{2} \) |
| 43 | \( 1 + 11.2iT - 43T^{2} \) |
| 47 | \( 1 + 2.82iT - 47T^{2} \) |
| 53 | \( 1 + 3.74iT - 53T^{2} \) |
| 59 | \( 1 - 9.89iT - 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 - 11.2iT - 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 8.48iT - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 + 15.8T + 89T^{2} \) |
| 97 | \( 1 - 5.29T + 97T^{2} \) |
| 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
−8.714442192210945473303239240909, −7.57549376587493863105127714052, −7.03926197964477468445764747766, −6.52859994140103623037243579082, −5.76053586299384739307858515631, −4.36056076818640365877345310778, −4.01579907670222613696680125774, −2.41651807437062993168512117246, −1.92544736014807106412119118707, −0.34120143828034444146307326355,
1.19868114107053519871606312506, 3.05279583462407334606070378734, 3.26751638191144619092672427188, 4.40977289880244070329385667335, 5.17166186363546572983278878038, 6.16465048817580451799729297717, 6.61519194247851451219562363802, 7.82200979741201191332987939111, 8.438793526127440781446386980236, 9.307433132896660491790845234829
