| L(s) = 1 | + (−2.78 + 0.491i)2-s + (−6.42 + 6.29i)3-s + (7.51 − 2.73i)4-s + (−15.4 + 18.4i)5-s + (14.8 − 20.7i)6-s + (5.84 + 2.12i)7-s + (−19.5 + 11.3i)8-s + (1.65 − 80.9i)9-s + (34.0 − 58.9i)10-s + (−49.4 − 58.9i)11-s + (−31.0 + 64.9i)12-s + (38.9 − 220. i)13-s + (−17.3 − 3.05i)14-s + (−16.6 − 216. i)15-s + (49.0 − 41.1i)16-s + (−99.8 − 57.6i)17-s + ⋯ |
| L(s) = 1 | + (−0.696 + 0.122i)2-s + (−0.714 + 0.699i)3-s + (0.469 − 0.171i)4-s + (−0.619 + 0.737i)5-s + (0.411 − 0.575i)6-s + (0.119 + 0.0433i)7-s + (−0.306 + 0.176i)8-s + (0.0203 − 0.999i)9-s + (0.340 − 0.589i)10-s + (−0.408 − 0.487i)11-s + (−0.215 + 0.450i)12-s + (0.230 − 1.30i)13-s + (−0.0883 − 0.0155i)14-s + (−0.0741 − 0.960i)15-s + (0.191 − 0.160i)16-s + (−0.345 − 0.199i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.232 + 0.972i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.232 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.123128 - 0.155957i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.123128 - 0.155957i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.78 - 0.491i)T \) |
| 3 | \( 1 + (6.42 - 6.29i)T \) |
| good | 5 | \( 1 + (15.4 - 18.4i)T + (-108. - 615. i)T^{2} \) |
| 7 | \( 1 + (-5.84 - 2.12i)T + (1.83e3 + 1.54e3i)T^{2} \) |
| 11 | \( 1 + (49.4 + 58.9i)T + (-2.54e3 + 1.44e4i)T^{2} \) |
| 13 | \( 1 + (-38.9 + 220. i)T + (-2.68e4 - 9.76e3i)T^{2} \) |
| 17 | \( 1 + (99.8 + 57.6i)T + (4.17e4 + 7.23e4i)T^{2} \) |
| 19 | \( 1 + (176. + 305. i)T + (-6.51e4 + 1.12e5i)T^{2} \) |
| 23 | \( 1 + (-102. - 281. i)T + (-2.14e5 + 1.79e5i)T^{2} \) |
| 29 | \( 1 + (893. - 157. i)T + (6.64e5 - 2.41e5i)T^{2} \) |
| 31 | \( 1 + (-100. + 36.4i)T + (7.07e5 - 5.93e5i)T^{2} \) |
| 37 | \( 1 + (-918. + 1.59e3i)T + (-9.37e5 - 1.62e6i)T^{2} \) |
| 41 | \( 1 + (2.28e3 + 402. i)T + (2.65e6 + 9.66e5i)T^{2} \) |
| 43 | \( 1 + (1.82e3 - 1.53e3i)T + (5.93e5 - 3.36e6i)T^{2} \) |
| 47 | \( 1 + (623. - 1.71e3i)T + (-3.73e6 - 3.13e6i)T^{2} \) |
| 53 | \( 1 + 5.60e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + (-2.69e3 + 3.20e3i)T + (-2.10e6 - 1.19e7i)T^{2} \) |
| 61 | \( 1 + (5.80e3 + 2.11e3i)T + (1.06e7 + 8.89e6i)T^{2} \) |
| 67 | \( 1 + (576. - 3.26e3i)T + (-1.89e7 - 6.89e6i)T^{2} \) |
| 71 | \( 1 + (1.90e3 + 1.10e3i)T + (1.27e7 + 2.20e7i)T^{2} \) |
| 73 | \( 1 + (-3.15e3 - 5.45e3i)T + (-1.41e7 + 2.45e7i)T^{2} \) |
| 79 | \( 1 + (-1.96e3 - 1.11e4i)T + (-3.66e7 + 1.33e7i)T^{2} \) |
| 83 | \( 1 + (-4.75e3 + 838. i)T + (4.45e7 - 1.62e7i)T^{2} \) |
| 89 | \( 1 + (6.38e3 - 3.68e3i)T + (3.13e7 - 5.43e7i)T^{2} \) |
| 97 | \( 1 + (7.05e3 - 5.91e3i)T + (1.53e7 - 8.71e7i)T^{2} \) |
| 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
−14.90712685401724793675877798392, −12.96945678405367508431702783011, −11.29757298190028372870512979596, −10.95139607723347407432261452848, −9.692098033062548004896320920777, −8.215071986096969654609940719620, −6.82438072182564283523612274132, −5.35850288529119141823609163617, −3.32568804485611505396279523959, −0.15806591492191820828046198027,
1.68940459502242778653758583036, 4.56898074604509510680117136456, 6.40543258055737186176709428425, 7.69112506278038659384331264380, 8.757407034206117882546504338761, 10.38066439848481901781017277846, 11.61389439454561428778326490865, 12.31552461024704176304722114555, 13.47658712368041297816346909642, 15.19256519943881743792738943481
