| L(s) = 1 | + (−8.30 − 3.02i)3-s + (2.68 + 15.2i)5-s + (13.2 − 22.9i)7-s + (39.1 + 32.8i)9-s + (17.8 + 30.8i)11-s + (81.8 − 29.7i)13-s + (23.7 − 134. i)15-s + (−60.6 + 50.8i)17-s + (55.9 + 61.0i)19-s + (−179. + 150. i)21-s + (−0.619 + 3.51i)23-s + (−107. + 39.1i)25-s + (−106. − 184. i)27-s + (−14.1 − 11.8i)29-s + (35.4 − 61.3i)31-s + ⋯ |
| L(s) = 1 | + (−1.59 − 0.581i)3-s + (0.240 + 1.36i)5-s + (0.714 − 1.23i)7-s + (1.44 + 1.21i)9-s + (0.488 + 0.845i)11-s + (1.74 − 0.635i)13-s + (0.408 − 2.31i)15-s + (−0.864 + 0.725i)17-s + (0.676 + 0.736i)19-s + (−1.86 + 1.56i)21-s + (−0.00561 + 0.0318i)23-s + (−0.860 + 0.313i)25-s + (−0.757 − 1.31i)27-s + (−0.0905 − 0.0759i)29-s + (0.205 − 0.355i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.202i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.979 - 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.04047 + 0.106365i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.04047 + 0.106365i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-55.9 - 61.0i)T \) |
| good | 3 | \( 1 + (8.30 + 3.02i)T + (20.6 + 17.3i)T^{2} \) |
| 5 | \( 1 + (-2.68 - 15.2i)T + (-117. + 42.7i)T^{2} \) |
| 7 | \( 1 + (-13.2 + 22.9i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-17.8 - 30.8i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-81.8 + 29.7i)T + (1.68e3 - 1.41e3i)T^{2} \) |
| 17 | \( 1 + (60.6 - 50.8i)T + (853. - 4.83e3i)T^{2} \) |
| 23 | \( 1 + (0.619 - 3.51i)T + (-1.14e4 - 4.16e3i)T^{2} \) |
| 29 | \( 1 + (14.1 + 11.8i)T + (4.23e3 + 2.40e4i)T^{2} \) |
| 31 | \( 1 + (-35.4 + 61.3i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 - 218.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-234. - 85.2i)T + (5.27e4 + 4.43e4i)T^{2} \) |
| 43 | \( 1 + (2.56 + 14.5i)T + (-7.47e4 + 2.71e4i)T^{2} \) |
| 47 | \( 1 + (16.4 + 13.8i)T + (1.80e4 + 1.02e5i)T^{2} \) |
| 53 | \( 1 + (-58.9 + 334. i)T + (-1.39e5 - 5.09e4i)T^{2} \) |
| 59 | \( 1 + (215. - 180. i)T + (3.56e4 - 2.02e5i)T^{2} \) |
| 61 | \( 1 + (85.9 - 487. i)T + (-2.13e5 - 7.76e4i)T^{2} \) |
| 67 | \( 1 + (35.8 + 30.0i)T + (5.22e4 + 2.96e5i)T^{2} \) |
| 71 | \( 1 + (77.7 + 441. i)T + (-3.36e5 + 1.22e5i)T^{2} \) |
| 73 | \( 1 + (-610. - 222. i)T + (2.98e5 + 2.50e5i)T^{2} \) |
| 79 | \( 1 + (946. + 344. i)T + (3.77e5 + 3.16e5i)T^{2} \) |
| 83 | \( 1 + (136. - 236. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (891. - 324. i)T + (5.40e5 - 4.53e5i)T^{2} \) |
| 97 | \( 1 + (616. - 517. i)T + (1.58e5 - 8.98e5i)T^{2} \) |
| 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
−13.89309713658610287774123026369, −12.94653381575101211808641390973, −11.47664205165024068514103265634, −10.90239616432703414516099193772, −10.22745344481714649457786594093, −7.74744110355889855686287822219, −6.75538535854242682855938455515, −5.93967987426014278526064544192, −4.12195074988732953693798979945, −1.32831240851431907027822593953,
1.05667066418766449517498370862, 4.45509267425948583174834214075, 5.42341721750192615489057805301, 6.25329367760704057442860039655, 8.684877195953189637775977411277, 9.268554078604504494472428218676, 11.24225353643464899906678380932, 11.44921625526252053238078591227, 12.59054754179757341138870063701, 13.79194411673479741987739324326
