| L(s) = 1 | + (−3.86 − 1.03i)2-s + (13.6 − 2.40i)3-s + (13.8 + 8.01i)4-s + (29.8 − 25.0i)5-s + (−55.1 − 4.86i)6-s + (54.4 + 31.4i)7-s + (−45.1 − 45.3i)8-s + (104. − 37.8i)9-s + (−141. + 65.8i)10-s + (−116. + 67.2i)11-s + (208. + 76.0i)12-s + (−36.2 + 205. i)13-s + (−177. − 177. i)14-s + (347. − 413. i)15-s + (127. + 222. i)16-s + (−331. − 120. i)17-s + ⋯ |
| L(s) = 1 | + (−0.965 − 0.259i)2-s + (1.51 − 0.267i)3-s + (0.865 + 0.501i)4-s + (1.19 − 1.00i)5-s + (−1.53 − 0.135i)6-s + (1.11 + 0.641i)7-s + (−0.705 − 0.708i)8-s + (1.28 − 0.467i)9-s + (−1.41 + 0.658i)10-s + (−0.962 + 0.555i)11-s + (1.44 + 0.528i)12-s + (−0.214 + 1.21i)13-s + (−0.906 − 0.908i)14-s + (1.54 − 1.83i)15-s + (0.497 + 0.867i)16-s + (−1.14 − 0.417i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.810 + 0.585i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.810 + 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.06284 - 0.666465i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.06284 - 0.666465i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.86 + 1.03i)T \) |
| 19 | \( 1 + (-248. + 261. i)T \) |
| good | 3 | \( 1 + (-13.6 + 2.40i)T + (76.1 - 27.7i)T^{2} \) |
| 5 | \( 1 + (-29.8 + 25.0i)T + (108. - 615. i)T^{2} \) |
| 7 | \( 1 + (-54.4 - 31.4i)T + (1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (116. - 67.2i)T + (7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (36.2 - 205. i)T + (-2.68e4 - 9.76e3i)T^{2} \) |
| 17 | \( 1 + (331. + 120. i)T + (6.39e4 + 5.36e4i)T^{2} \) |
| 23 | \( 1 + (-246. + 294. i)T + (-4.85e4 - 2.75e5i)T^{2} \) |
| 29 | \( 1 + (739. - 269. i)T + (5.41e5 - 4.54e5i)T^{2} \) |
| 31 | \( 1 + (149. + 86.5i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + 1.11e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + (-308. - 1.74e3i)T + (-2.65e6 + 9.66e5i)T^{2} \) |
| 43 | \( 1 + (-547. - 652. i)T + (-5.93e5 + 3.36e6i)T^{2} \) |
| 47 | \( 1 + (1.40e3 + 3.86e3i)T + (-3.73e6 + 3.13e6i)T^{2} \) |
| 53 | \( 1 + (-361. - 303. i)T + (1.37e6 + 7.77e6i)T^{2} \) |
| 59 | \( 1 + (-818. + 2.24e3i)T + (-9.28e6 - 7.78e6i)T^{2} \) |
| 61 | \( 1 + (2.12e3 + 1.78e3i)T + (2.40e6 + 1.36e7i)T^{2} \) |
| 67 | \( 1 + (411. + 1.13e3i)T + (-1.54e7 + 1.29e7i)T^{2} \) |
| 71 | \( 1 + (-4.31e3 - 5.13e3i)T + (-4.41e6 + 2.50e7i)T^{2} \) |
| 73 | \( 1 + (-562. - 3.19e3i)T + (-2.66e7 + 9.71e6i)T^{2} \) |
| 79 | \( 1 + (3.47e3 - 613. i)T + (3.66e7 - 1.33e7i)T^{2} \) |
| 83 | \( 1 + (-147. - 85.1i)T + (2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + (243. - 1.38e3i)T + (-5.89e7 - 2.14e7i)T^{2} \) |
| 97 | \( 1 + (-352. - 128. i)T + (6.78e7 + 5.69e7i)T^{2} \) |
| 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
−13.56615112389647916863035134827, −12.75672803751557158924528757210, −11.35769249579176162305982736710, −9.686058351192720045657350220340, −9.017237835908390072080020380908, −8.374868716513712805795052659812, −7.08569333205327421197386181549, −4.94995125434617888031950338270, −2.38321350774554583381859376484, −1.76705723126093842529686617537,
1.89083245775867245442631814269, 3.03755132900618988683449195975, 5.62196654714850757192981625491, 7.38871613078090871948798555367, 8.104768355335562548300792837846, 9.300752047222294186534225362903, 10.39659128118034983002608516030, 10.88857522361278570344326446143, 13.35174365848353972336705107518, 14.13863680154090026639826110915
