L(s) = 1 | + (−0.608 + 0.793i)2-s + (−0.258 − 0.965i)4-s + (−1.05 + 0.357i)5-s + (0.923 + 0.382i)8-s + (0.991 − 0.130i)9-s + (0.357 − 1.05i)10-s + (−1 + i)13-s + (−0.866 + 0.499i)16-s + (0.130 − 0.991i)17-s + (−0.499 + 0.866i)18-s + (0.617 + 0.923i)20-s + (0.186 − 0.142i)25-s + (−0.184 − 1.40i)26-s + (−0.216 + 1.08i)29-s + (0.130 − 0.991i)32-s + ⋯ |
L(s) = 1 | + (−0.608 + 0.793i)2-s + (−0.258 − 0.965i)4-s + (−1.05 + 0.357i)5-s + (0.923 + 0.382i)8-s + (0.991 − 0.130i)9-s + (0.357 − 1.05i)10-s + (−1 + i)13-s + (−0.866 + 0.499i)16-s + (0.130 − 0.991i)17-s + (−0.499 + 0.866i)18-s + (0.617 + 0.923i)20-s + (0.186 − 0.142i)25-s + (−0.184 − 1.40i)26-s + (−0.216 + 1.08i)29-s + (0.130 − 0.991i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.666 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.666 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5789406510\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5789406510\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.608 - 0.793i)T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + (-0.130 + 0.991i)T \) |
good | 3 | \( 1 + (-0.991 + 0.130i)T^{2} \) |
| 5 | \( 1 + (1.05 - 0.357i)T + (0.793 - 0.608i)T^{2} \) |
| 11 | \( 1 + (0.608 - 0.793i)T^{2} \) |
| 13 | \( 1 + (1 - i)T - iT^{2} \) |
| 19 | \( 1 + (-0.258 - 0.965i)T^{2} \) |
| 23 | \( 1 + (0.991 + 0.130i)T^{2} \) |
| 29 | \( 1 + (0.216 - 1.08i)T + (-0.923 - 0.382i)T^{2} \) |
| 31 | \( 1 + (0.991 - 0.130i)T^{2} \) |
| 37 | \( 1 + (-1.49 - 0.735i)T + (0.608 + 0.793i)T^{2} \) |
| 41 | \( 1 + (-0.0761 - 0.382i)T + (-0.923 + 0.382i)T^{2} \) |
| 43 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 47 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 + (1.83 + 0.241i)T + (0.965 + 0.258i)T^{2} \) |
| 59 | \( 1 + (0.258 - 0.965i)T^{2} \) |
| 61 | \( 1 + (-1.29 - 1.47i)T + (-0.130 + 0.991i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.382 + 0.923i)T^{2} \) |
| 73 | \( 1 + (-1.25 - 1.09i)T + (0.130 + 0.991i)T^{2} \) |
| 79 | \( 1 + (-0.991 - 0.130i)T^{2} \) |
| 83 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 89 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 97 | \( 1 + (1.63 + 0.324i)T + (0.923 + 0.382i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.151304463903882382231514920697, −8.066816174828938853669735454700, −7.56490133324294757145032279517, −6.95797478193490372036708363066, −6.52341552466010246447804689388, −5.21863553421104417821659350050, −4.58561838389045615742111246947, −3.85448391008407747875979987941, −2.53821088947863823402635413079, −1.22464392656625361570815943220,
0.48229656448719361807815991016, 1.78608899108506902513280801517, 2.83950699195700006263559722249, 3.89796419552825215956496036629, 4.31072007649998197190173650290, 5.24125893460928990671670706445, 6.50362921899892937668761051591, 7.54584185306729138932501855914, 7.85523868428021230777218244580, 8.364944329986031289272731852377