| L(s) = 1 | + (−4.13 + 11.3i)3-s + (−6.25 − 35.4i)5-s + (−25.8 + 44.8i)7-s + (−50.1 − 42.1i)9-s + (−31.3 − 54.3i)11-s + (−109. − 301. i)13-s + (429. + 75.6i)15-s + (342. − 287. i)17-s + (−162. − 322. i)19-s + (−402. − 479. i)21-s + (−90.4 + 513. i)23-s + (−630. + 229. i)25-s + (−162. + 93.7i)27-s + (−524. + 625. i)29-s + (−1.07e3 − 619. i)31-s + ⋯ |
| L(s) = 1 | + (−0.459 + 1.26i)3-s + (−0.250 − 1.41i)5-s + (−0.527 + 0.914i)7-s + (−0.619 − 0.519i)9-s + (−0.259 − 0.448i)11-s + (−0.649 − 1.78i)13-s + (1.90 + 0.336i)15-s + (1.18 − 0.993i)17-s + (−0.451 − 0.892i)19-s + (−0.912 − 1.08i)21-s + (−0.171 + 0.970i)23-s + (−1.00 + 0.367i)25-s + (−0.222 + 0.128i)27-s + (−0.624 + 0.743i)29-s + (−1.11 − 0.644i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.228 + 0.973i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.228 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.323840 - 0.408676i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.323840 - 0.408676i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (162. + 322. i)T \) |
| good | 3 | \( 1 + (4.13 - 11.3i)T + (-62.0 - 52.0i)T^{2} \) |
| 5 | \( 1 + (6.25 + 35.4i)T + (-587. + 213. i)T^{2} \) |
| 7 | \( 1 + (25.8 - 44.8i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (31.3 + 54.3i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (109. + 301. i)T + (-2.18e4 + 1.83e4i)T^{2} \) |
| 17 | \( 1 + (-342. + 287. i)T + (1.45e4 - 8.22e4i)T^{2} \) |
| 23 | \( 1 + (90.4 - 513. i)T + (-2.62e5 - 9.57e4i)T^{2} \) |
| 29 | \( 1 + (524. - 625. i)T + (-1.22e5 - 6.96e5i)T^{2} \) |
| 31 | \( 1 + (1.07e3 + 619. i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 - 532. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (355. - 975. i)T + (-2.16e6 - 1.81e6i)T^{2} \) |
| 43 | \( 1 + (79.7 + 452. i)T + (-3.21e6 + 1.16e6i)T^{2} \) |
| 47 | \( 1 + (2.40e3 + 2.02e3i)T + (8.47e5 + 4.80e6i)T^{2} \) |
| 53 | \( 1 + (1.79e3 + 316. i)T + (7.41e6 + 2.69e6i)T^{2} \) |
| 59 | \( 1 + (365. + 435. i)T + (-2.10e6 + 1.19e7i)T^{2} \) |
| 61 | \( 1 + (-93.8 + 531. i)T + (-1.30e7 - 4.73e6i)T^{2} \) |
| 67 | \( 1 + (-4.28e3 + 5.10e3i)T + (-3.49e6 - 1.98e7i)T^{2} \) |
| 71 | \( 1 + (1.60e3 - 283. i)T + (2.38e7 - 8.69e6i)T^{2} \) |
| 73 | \( 1 + (-7.74e3 - 2.81e3i)T + (2.17e7 + 1.82e7i)T^{2} \) |
| 79 | \( 1 + (2.37e3 - 6.53e3i)T + (-2.98e7 - 2.50e7i)T^{2} \) |
| 83 | \( 1 + (-3.68e3 + 6.38e3i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (-2.15e3 - 5.91e3i)T + (-4.80e7 + 4.03e7i)T^{2} \) |
| 97 | \( 1 + (2.69e3 + 3.21e3i)T + (-1.53e7 + 8.71e7i)T^{2} \) |
| 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
−13.13576393870997079561608955769, −12.38387466538524094465920125533, −11.23259098799754467483623050328, −9.871137288993780058571322060206, −9.185770591102076091129957506749, −7.915306519299534858133254390644, −5.41512423173723278766918132278, −5.15617256230641321021487876746, −3.31464952289697613864227987152, −0.27326469509585558284806900690,
1.92589239073889974879264585418, 3.84006739148062947429184062603, 6.26341413895563335899950032391, 6.97351108703021916506919369816, 7.72915713586776681881714692385, 9.888067200975890617183806644512, 10.85600561198052065103043727401, 12.01915121548358099508316071609, 12.84941997727606090653760127813, 14.17646552363305738162582053596
