| 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
−14.17646552363305738162582053596, −12.84941997727606090653760127813, −12.01915121548358099508316071609, −10.85600561198052065103043727401, −9.888067200975890617183806644512, −7.72915713586776681881714692385, −6.97351108703021916506919369816, −6.26341413895563335899950032391, −3.84006739148062947429184062603, −1.92589239073889974879264585418,
0.27326469509585558284806900690, 3.31464952289697613864227987152, 5.15617256230641321021487876746, 5.41512423173723278766918132278, 7.915306519299534858133254390644, 9.185770591102076091129957506749, 9.871137288993780058571322060206, 11.23259098799754467483623050328, 12.38387466538524094465920125533, 13.13576393870997079561608955769
