| L(s) = 1 | + (−0.284 − 1.97i)2-s + (−1.24 + 2.72i)3-s + (−3.83 + 1.12i)4-s + (1.48 − 1.70i)5-s + (5.75 + 1.69i)6-s + (3.78 − 2.43i)7-s + (3.32 + 7.27i)8-s + (−5.89 − 6.80i)9-s + (−3.80 − 2.44i)10-s + (−2.10 + 14.6i)11-s + (1.70 − 11.8i)12-s + (57.7 + 37.0i)13-s + (−5.89 − 6.80i)14-s + (2.81 + 6.16i)15-s + (13.4 − 8.65i)16-s + (65.6 + 19.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 − 0.699i)2-s + (−0.239 + 0.525i)3-s + (−0.479 + 0.140i)4-s + (0.132 − 0.152i)5-s + (0.391 + 0.115i)6-s + (0.204 − 0.131i)7-s + (0.146 + 0.321i)8-s + (−0.218 − 0.251i)9-s + (−0.120 − 0.0772i)10-s + (−0.0577 + 0.401i)11-s + (0.0410 − 0.285i)12-s + (1.23 + 0.791i)13-s + (−0.112 − 0.129i)14-s + (0.0484 + 0.106i)15-s + (0.210 − 0.135i)16-s + (0.937 + 0.275i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.124i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.992 + 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.49647 - 0.0933519i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.49647 - 0.0933519i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.284 + 1.97i)T \) |
| 3 | \( 1 + (1.24 - 2.72i)T \) |
| 23 | \( 1 + (-79.2 + 76.7i)T \) |
| good | 5 | \( 1 + (-1.48 + 1.70i)T + (-17.7 - 123. i)T^{2} \) |
| 7 | \( 1 + (-3.78 + 2.43i)T + (142. - 312. i)T^{2} \) |
| 11 | \( 1 + (2.10 - 14.6i)T + (-1.27e3 - 374. i)T^{2} \) |
| 13 | \( 1 + (-57.7 - 37.0i)T + (912. + 1.99e3i)T^{2} \) |
| 17 | \( 1 + (-65.6 - 19.2i)T + (4.13e3 + 2.65e3i)T^{2} \) |
| 19 | \( 1 + (-109. + 32.1i)T + (5.77e3 - 3.70e3i)T^{2} \) |
| 29 | \( 1 + (154. + 45.5i)T + (2.05e4 + 1.31e4i)T^{2} \) |
| 31 | \( 1 + (-19.3 - 42.4i)T + (-1.95e4 + 2.25e4i)T^{2} \) |
| 37 | \( 1 + (-46.5 - 53.7i)T + (-7.20e3 + 5.01e4i)T^{2} \) |
| 41 | \( 1 + (-179. + 207. i)T + (-9.80e3 - 6.82e4i)T^{2} \) |
| 43 | \( 1 + (52.4 - 114. i)T + (-5.20e4 - 6.00e4i)T^{2} \) |
| 47 | \( 1 - 290.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (330. - 212. i)T + (6.18e4 - 1.35e5i)T^{2} \) |
| 59 | \( 1 + (652. + 419. i)T + (8.53e4 + 1.86e5i)T^{2} \) |
| 61 | \( 1 + (-271. - 595. i)T + (-1.48e5 + 1.71e5i)T^{2} \) |
| 67 | \( 1 + (23.8 + 166. i)T + (-2.88e5 + 8.47e4i)T^{2} \) |
| 71 | \( 1 + (29.3 + 204. i)T + (-3.43e5 + 1.00e5i)T^{2} \) |
| 73 | \( 1 + (540. - 158. i)T + (3.27e5 - 2.10e5i)T^{2} \) |
| 79 | \( 1 + (188. + 120. i)T + (2.04e5 + 4.48e5i)T^{2} \) |
| 83 | \( 1 + (14.5 + 16.8i)T + (-8.13e4 + 5.65e5i)T^{2} \) |
| 89 | \( 1 + (-326. + 714. i)T + (-4.61e5 - 5.32e5i)T^{2} \) |
| 97 | \( 1 + (-510. + 588. i)T + (-1.29e5 - 9.03e5i)T^{2} \) |
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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
−12.54816045287562199092008418673, −11.46676565781799154464145088469, −10.79911773231155160070434470641, −9.623549323812977964776296437988, −8.887357381486600819504136962116, −7.45410422395929023079432515138, −5.80855146279480050603922268277, −4.56437885008107553712756927560, −3.30092060775115812482202514295, −1.29752670583523981500044593559,
1.04633693418780446304148221865, 3.35043145529751669519744366607, 5.32828142393674350106489515742, 6.07940171478349218324233831878, 7.43549659935460060592661595096, 8.223405488936517058563400024565, 9.464182508185774560843601144824, 10.71716853250663102382330004703, 11.74693318878548642557162903647, 12.95619884011577193857110167877
