| L(s) = 1 | + (8.12 − 24.9i)3-s + (−45.3 + 32.9i)5-s + (23.8 + 73.4i)7-s + (−362. − 263. i)9-s + (−391. − 87.7i)11-s + (−517. − 376. i)13-s + (454. + 1.40e3i)15-s + (−405. + 294. i)17-s + (−784. + 2.41e3i)19-s + 2.02e3·21-s + 2.71e3·23-s + (4.24 − 13.0i)25-s + (−4.34e3 + 3.15e3i)27-s + (750. + 2.30e3i)29-s + (−5.14e3 − 3.74e3i)31-s + ⋯ |
| L(s) = 1 | + (0.520 − 1.60i)3-s + (−0.810 + 0.589i)5-s + (0.184 + 0.566i)7-s + (−1.48 − 1.08i)9-s + (−0.975 − 0.218i)11-s + (−0.849 − 0.617i)13-s + (0.522 + 1.60i)15-s + (−0.340 + 0.247i)17-s + (−0.498 + 1.53i)19-s + 1.00·21-s + 1.06·23-s + (0.00135 − 0.00417i)25-s + (−1.14 + 0.833i)27-s + (0.165 + 0.509i)29-s + (−0.962 − 0.699i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.678 - 0.734i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0566032 + 0.129329i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0566032 + 0.129329i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + (391. + 87.7i)T \) |
| good | 3 | \( 1 + (-8.12 + 24.9i)T + (-196. - 142. i)T^{2} \) |
| 5 | \( 1 + (45.3 - 32.9i)T + (965. - 2.97e3i)T^{2} \) |
| 7 | \( 1 + (-23.8 - 73.4i)T + (-1.35e4 + 9.87e3i)T^{2} \) |
| 13 | \( 1 + (517. + 376. i)T + (1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (405. - 294. i)T + (4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (784. - 2.41e3i)T + (-2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 - 2.71e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + (-750. - 2.30e3i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (5.14e3 + 3.74e3i)T + (8.84e6 + 2.72e7i)T^{2} \) |
| 37 | \( 1 + (1.40e3 + 4.32e3i)T + (-5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (-519. + 1.59e3i)T + (-9.37e7 - 6.80e7i)T^{2} \) |
| 43 | \( 1 + 1.83e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-9.06e3 + 2.78e4i)T + (-1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (1.82e4 + 1.32e4i)T + (1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-8.88e3 - 2.73e4i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (3.40e3 - 2.47e3i)T + (2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 + 5.80e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (5.48e3 - 3.98e3i)T + (5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (2.35e4 + 7.23e4i)T + (-1.67e9 + 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-2.47e4 - 1.79e4i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (6.73e4 - 4.89e4i)T + (1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 - 337.T + 5.58e9T^{2} \) |
| 97 | \( 1 + (1.08e5 + 7.86e4i)T + (2.65e9 + 8.16e9i)T^{2} \) |
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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
−12.59173564770046536141878381666, −11.77391690697701155120353423589, −10.51488678889143864641215221832, −8.642626011839606183486773157393, −7.79463311640551560166936870044, −7.05810253509184550727415672546, −5.59349725029573681811929667052, −3.23922554523837044738828942180, −2.02916998347618740046893548654, −0.05056494678035260321289440613,
2.88582358132626936409288066861, 4.47346619297873953534210600627, 4.83708930855897141949639549640, 7.30596275270851914646619559520, 8.559470257112027520569834698922, 9.432707210188969873461390152002, 10.56610792517740712190999419985, 11.39695826429126281677678154548, 12.87628810554672596740066384492, 14.10247036209719952396789231969
