| L(s) = 1 | + (2.66 + 4.46i)3-s + (−12.8 − 10.7i)5-s + (−20.8 − 7.57i)7-s + (−12.8 + 23.7i)9-s + (−41.5 + 34.8i)11-s + (12.2 − 69.4i)13-s + (13.8 − 85.7i)15-s + (17.7 − 30.7i)17-s + (37.9 + 65.7i)19-s + (−21.5 − 112. i)21-s + (−161. + 58.6i)23-s + (26.8 + 152. i)25-s + (−140. + 5.97i)27-s + (−25.1 − 142. i)29-s + (160. − 58.2i)31-s + ⋯ |
| L(s) = 1 | + (0.512 + 0.858i)3-s + (−1.14 − 0.960i)5-s + (−1.12 − 0.408i)7-s + (−0.475 + 0.879i)9-s + (−1.13 + 0.955i)11-s + (0.261 − 1.48i)13-s + (0.238 − 1.47i)15-s + (0.252 − 0.437i)17-s + (0.458 + 0.793i)19-s + (−0.224 − 1.17i)21-s + (−1.46 + 0.531i)23-s + (0.214 + 1.21i)25-s + (−0.999 + 0.0425i)27-s + (−0.161 − 0.913i)29-s + (0.927 − 0.337i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.434i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.900 + 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0255686 - 0.111753i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0255686 - 0.111753i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.66 - 4.46i)T \) |
| good | 5 | \( 1 + (12.8 + 10.7i)T + (21.7 + 123. i)T^{2} \) |
| 7 | \( 1 + (20.8 + 7.57i)T + (262. + 220. i)T^{2} \) |
| 11 | \( 1 + (41.5 - 34.8i)T + (231. - 1.31e3i)T^{2} \) |
| 13 | \( 1 + (-12.2 + 69.4i)T + (-2.06e3 - 751. i)T^{2} \) |
| 17 | \( 1 + (-17.7 + 30.7i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-37.9 - 65.7i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (161. - 58.6i)T + (9.32e3 - 7.82e3i)T^{2} \) |
| 29 | \( 1 + (25.1 + 142. i)T + (-2.29e4 + 8.34e3i)T^{2} \) |
| 31 | \( 1 + (-160. + 58.2i)T + (2.28e4 - 1.91e4i)T^{2} \) |
| 37 | \( 1 + (180. - 313. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (24.4 - 138. i)T + (-6.47e4 - 2.35e4i)T^{2} \) |
| 43 | \( 1 + (-269. + 226. i)T + (1.38e4 - 7.82e4i)T^{2} \) |
| 47 | \( 1 + (74.2 + 27.0i)T + (7.95e4 + 6.67e4i)T^{2} \) |
| 53 | \( 1 + 195.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (199. + 167. i)T + (3.56e4 + 2.02e5i)T^{2} \) |
| 61 | \( 1 + (-347. - 126. i)T + (1.73e5 + 1.45e5i)T^{2} \) |
| 67 | \( 1 + (13.6 - 77.4i)T + (-2.82e5 - 1.02e5i)T^{2} \) |
| 71 | \( 1 + (365. - 633. i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + (361. + 626. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-52.6 - 298. i)T + (-4.63e5 + 1.68e5i)T^{2} \) |
| 83 | \( 1 + (-8.20 - 46.5i)T + (-5.37e5 + 1.95e5i)T^{2} \) |
| 89 | \( 1 + (340. + 589. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-426. + 358. i)T + (1.58e5 - 8.98e5i)T^{2} \) |
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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
−12.81504176615670175139239792342, −11.85963565969833979944453626498, −10.23875761547098544892563025996, −9.804549102612572258332063036479, −8.197644109473158596789012607564, −7.70385869284075100613761676002, −5.49875548386592945427615856719, −4.25825142682932179725610987498, −3.14062252044821812784274928929, −0.05605312044804790893152907321,
2.67367511052098168501207152144, 3.67747654164140928248709059929, 6.13429258695043977969656018504, 7.02963449174480722880592998556, 8.072153593349021481977289944117, 9.144721986668449298445540779323, 10.68435466084987824429580246968, 11.72621135449829442478232507751, 12.58846092198793833379198718352, 13.72845957597840060334242720185
