L(s) = 1 | + (−1.05 + 0.945i)2-s + 1.57i·3-s + (0.210 − 1.98i)4-s + i·5-s + (−1.49 − 1.65i)6-s + 1.14·7-s + (1.66 + 2.28i)8-s + 0.509·9-s + (−0.945 − 1.05i)10-s + 5.86·11-s + (3.13 + 0.331i)12-s − 2.69·13-s + (−1.20 + 1.08i)14-s − 1.57·15-s + (−3.91 − 0.836i)16-s + 0.337i·17-s + ⋯ |
L(s) = 1 | + (−0.743 + 0.668i)2-s + 0.911i·3-s + (0.105 − 0.994i)4-s + 0.447i·5-s + (−0.609 − 0.677i)6-s + 0.434·7-s + (0.587 + 0.809i)8-s + 0.169·9-s + (−0.299 − 0.332i)10-s + 1.76·11-s + (0.906 + 0.0958i)12-s − 0.747·13-s + (−0.322 + 0.290i)14-s − 0.407·15-s + (−0.977 − 0.209i)16-s + 0.0818i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.414 - 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.414 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.604394 + 0.939436i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.604394 + 0.939436i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.05 - 0.945i)T \) |
| 5 | \( 1 - iT \) |
| 23 | \( 1 + (-2.43 - 4.13i)T \) |
good | 3 | \( 1 - 1.57iT - 3T^{2} \) |
| 7 | \( 1 - 1.14T + 7T^{2} \) |
| 11 | \( 1 - 5.86T + 11T^{2} \) |
| 13 | \( 1 + 2.69T + 13T^{2} \) |
| 17 | \( 1 - 0.337iT - 17T^{2} \) |
| 19 | \( 1 - 4.49T + 19T^{2} \) |
| 29 | \( 1 + 9.68T + 29T^{2} \) |
| 31 | \( 1 + 4.80iT - 31T^{2} \) |
| 37 | \( 1 - 3.78iT - 37T^{2} \) |
| 41 | \( 1 - 7.24T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 + 2.80iT - 47T^{2} \) |
| 53 | \( 1 + 7.11iT - 53T^{2} \) |
| 59 | \( 1 - 1.70iT - 59T^{2} \) |
| 61 | \( 1 - 13.9iT - 61T^{2} \) |
| 67 | \( 1 - 6.00T + 67T^{2} \) |
| 71 | \( 1 - 3.87iT - 71T^{2} \) |
| 73 | \( 1 + 1.87T + 73T^{2} \) |
| 79 | \( 1 + 5.07T + 79T^{2} \) |
| 83 | \( 1 - 15.7T + 83T^{2} \) |
| 89 | \( 1 + 18.1iT - 89T^{2} \) |
| 97 | \( 1 + 14.5iT - 97T^{2} \) |
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
−11.34119586663589051960613106877, −10.03250920293547076085250580600, −9.597082626039868810752396471562, −8.891230126320300045885675955757, −7.56567946814496456925474835803, −6.95014112001168816823264476148, −5.75263261208760328190555223946, −4.73683340025282525968724450398, −3.61913401492532724160040393881, −1.60097650371928735192495688640,
1.04490826323822864806781222707, 1.98560572852114321225913681216, 3.62201609850464197986584813930, 4.81161034935123264400821313082, 6.50401642848244334033940780459, 7.28130019101076467587570359432, 8.045497173529924178278472875211, 9.167134943529877080792110285713, 9.604569995583280547791765859490, 10.93559489016714108816858798275