| L(s) = 1 | + (−1.87 − 0.684i)2-s + (2.43 − 4.59i)3-s + (3.06 + 2.57i)4-s + (1.31 − 7.46i)5-s + (−7.71 + 6.96i)6-s + (−2.90 + 2.43i)7-s + (−4.00 − 6.92i)8-s + (−15.1 − 22.3i)9-s + (−7.57 + 13.1i)10-s + (−8.73 − 49.5i)11-s + (19.2 − 7.80i)12-s + (−7.25 + 2.64i)13-s + (7.11 − 2.59i)14-s + (−31.0 − 24.1i)15-s + (2.77 + 15.7i)16-s + (17.4 − 30.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 − 0.241i)2-s + (0.468 − 0.883i)3-s + (0.383 + 0.321i)4-s + (0.117 − 0.667i)5-s + (−0.524 + 0.473i)6-s + (−0.156 + 0.131i)7-s + (−0.176 − 0.306i)8-s + (−0.561 − 0.827i)9-s + (−0.239 + 0.414i)10-s + (−0.239 − 1.35i)11-s + (0.463 − 0.187i)12-s + (−0.154 + 0.0563i)13-s + (0.135 − 0.0494i)14-s + (−0.534 − 0.416i)15-s + (0.0434 + 0.246i)16-s + (0.249 − 0.431i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.349 + 0.936i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.349 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.628433 - 0.905111i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.628433 - 0.905111i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.87 + 0.684i)T \) |
| 3 | \( 1 + (-2.43 + 4.59i)T \) |
| good | 5 | \( 1 + (-1.31 + 7.46i)T + (-117. - 42.7i)T^{2} \) |
| 7 | \( 1 + (2.90 - 2.43i)T + (59.5 - 337. i)T^{2} \) |
| 11 | \( 1 + (8.73 + 49.5i)T + (-1.25e3 + 455. i)T^{2} \) |
| 13 | \( 1 + (7.25 - 2.64i)T + (1.68e3 - 1.41e3i)T^{2} \) |
| 17 | \( 1 + (-17.4 + 30.2i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-36.2 - 62.8i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-116. - 97.6i)T + (2.11e3 + 1.19e4i)T^{2} \) |
| 29 | \( 1 + (60.4 + 21.9i)T + (1.86e4 + 1.56e4i)T^{2} \) |
| 31 | \( 1 + (-175. - 147. i)T + (5.17e3 + 2.93e4i)T^{2} \) |
| 37 | \( 1 + (-121. + 209. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-382. + 139. i)T + (5.27e4 - 4.43e4i)T^{2} \) |
| 43 | \( 1 + (62.2 + 353. i)T + (-7.47e4 + 2.71e4i)T^{2} \) |
| 47 | \( 1 + (385. - 323. i)T + (1.80e4 - 1.02e5i)T^{2} \) |
| 53 | \( 1 + 70.4T + 1.48e5T^{2} \) |
| 59 | \( 1 + (144. - 818. i)T + (-1.92e5 - 7.02e4i)T^{2} \) |
| 61 | \( 1 + (-540. + 453. i)T + (3.94e4 - 2.23e5i)T^{2} \) |
| 67 | \( 1 + (372. - 135. i)T + (2.30e5 - 1.93e5i)T^{2} \) |
| 71 | \( 1 + (78.6 - 136. i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + (152. + 264. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-747. - 272. i)T + (3.77e5 + 3.16e5i)T^{2} \) |
| 83 | \( 1 + (1.03e3 + 377. i)T + (4.38e5 + 3.67e5i)T^{2} \) |
| 89 | \( 1 + (109. + 189. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (113. + 644. i)T + (-8.57e5 + 3.12e5i)T^{2} \) |
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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
−14.29069849183535245774963604238, −13.23901156300881721381601073018, −12.27057144124569751774892977072, −11.12340608609175115261712730842, −9.403844729786272824577921898583, −8.536635413168451225270999014660, −7.39841884368838642520868738914, −5.77977294243426627100295054284, −3.07382562055579352503581285731, −1.02790748929053957743972254799,
2.71298154090567173752173334865, 4.77508912286393222615085389901, 6.73305519958166185091811101115, 8.041195550019268534889639501629, 9.492902816471050496783758839406, 10.21261022328019704879693951274, 11.29894099743464542751974625673, 13.07848180310416468309349221758, 14.68575927009037821027518735932, 15.04587218645266305330182488987
