| L(s) = 1 | + (−4.65 + 12.7i)3-s + (6.59 + 37.3i)5-s + (5.00 − 8.66i)7-s + (−79.9 − 67.0i)9-s + (−7.63 − 13.2i)11-s + (−22.1 − 60.9i)13-s + (−508. − 89.7i)15-s + (−348. + 292. i)17-s + (281. − 226. i)19-s + (87.5 + 104. i)21-s + (49.2 − 279. i)23-s + (−766. + 278. i)25-s + (275. − 158. i)27-s + (−472. + 562. i)29-s + (1.43e3 + 825. i)31-s + ⋯ |
| L(s) = 1 | + (−0.517 + 1.42i)3-s + (0.263 + 1.49i)5-s + (0.102 − 0.176i)7-s + (−0.986 − 0.828i)9-s + (−0.0631 − 0.109i)11-s + (−0.131 − 0.360i)13-s + (−2.26 − 0.398i)15-s + (−1.20 + 1.01i)17-s + (0.778 − 0.627i)19-s + (0.198 + 0.236i)21-s + (0.0931 − 0.528i)23-s + (−1.22 + 0.446i)25-s + (0.377 − 0.218i)27-s + (−0.561 + 0.669i)29-s + (1.48 + 0.859i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0208i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0208i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.0113965 + 1.09441i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0113965 + 1.09441i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-281. + 226. i)T \) |
| good | 3 | \( 1 + (4.65 - 12.7i)T + (-62.0 - 52.0i)T^{2} \) |
| 5 | \( 1 + (-6.59 - 37.3i)T + (-587. + 213. i)T^{2} \) |
| 7 | \( 1 + (-5.00 + 8.66i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (7.63 + 13.2i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (22.1 + 60.9i)T + (-2.18e4 + 1.83e4i)T^{2} \) |
| 17 | \( 1 + (348. - 292. i)T + (1.45e4 - 8.22e4i)T^{2} \) |
| 23 | \( 1 + (-49.2 + 279. i)T + (-2.62e5 - 9.57e4i)T^{2} \) |
| 29 | \( 1 + (472. - 562. i)T + (-1.22e5 - 6.96e5i)T^{2} \) |
| 31 | \( 1 + (-1.43e3 - 825. i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + 1.47e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (842. - 2.31e3i)T + (-2.16e6 - 1.81e6i)T^{2} \) |
| 43 | \( 1 + (329. + 1.86e3i)T + (-3.21e6 + 1.16e6i)T^{2} \) |
| 47 | \( 1 + (-985. - 827. i)T + (8.47e5 + 4.80e6i)T^{2} \) |
| 53 | \( 1 + (-3.11e3 - 549. i)T + (7.41e6 + 2.69e6i)T^{2} \) |
| 59 | \( 1 + (-938. - 1.11e3i)T + (-2.10e6 + 1.19e7i)T^{2} \) |
| 61 | \( 1 + (920. - 5.21e3i)T + (-1.30e7 - 4.73e6i)T^{2} \) |
| 67 | \( 1 + (4.06e3 - 4.84e3i)T + (-3.49e6 - 1.98e7i)T^{2} \) |
| 71 | \( 1 + (9.08e3 - 1.60e3i)T + (2.38e7 - 8.69e6i)T^{2} \) |
| 73 | \( 1 + (3.63e3 + 1.32e3i)T + (2.17e7 + 1.82e7i)T^{2} \) |
| 79 | \( 1 + (1.43e3 - 3.95e3i)T + (-2.98e7 - 2.50e7i)T^{2} \) |
| 83 | \( 1 + (-6.21e3 + 1.07e4i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (-4.13e3 - 1.13e4i)T + (-4.80e7 + 4.03e7i)T^{2} \) |
| 97 | \( 1 + (1.90e3 + 2.27e3i)T + (-1.53e7 + 8.71e7i)T^{2} \) |
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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
−14.67093745061315908503663205389, −13.44291260680767230032841053897, −11.64754354423140989756709006715, −10.59437478786967578598176000742, −10.36834068246534644821834464617, −8.936949508719686204837811609169, −7.07263654082708889470181704482, −5.83196871560183604890904480523, −4.34854039272836073712435032062, −2.93576973296331298412622297448,
0.59713940044553799632020441467, 1.92754594308693210972277858931, 4.79498963206766465375998566934, 5.97389159811509695414267302770, 7.30777010116439045169279678209, 8.468001887393485787815262481608, 9.620710013114885651902094530992, 11.65192214063342309728413070978, 12.07732760633665001709209166086, 13.32208094269705319411850722975
