| L(s) = 1 | + (5.99 − 1.05i)3-s + (34.0 − 28.5i)5-s + (−4.84 + 8.39i)7-s + (−41.2 + 15.0i)9-s + (−85.0 − 147. i)11-s + (264. + 46.5i)13-s + (173. − 207. i)15-s + (−15.8 − 5.78i)17-s + (360. + 1.73i)19-s + (−20.1 + 55.4i)21-s + (−237. − 199. i)23-s + (233. − 1.32e3i)25-s + (−658. + 380. i)27-s + (408. + 1.12e3i)29-s + (591. + 341. i)31-s + ⋯ |
| L(s) = 1 | + (0.666 − 0.117i)3-s + (1.36 − 1.14i)5-s + (−0.0988 + 0.171i)7-s + (−0.509 + 0.185i)9-s + (−0.703 − 1.21i)11-s + (1.56 + 0.275i)13-s + (0.772 − 0.920i)15-s + (−0.0549 − 0.0200i)17-s + (0.999 + 0.00479i)19-s + (−0.0457 + 0.125i)21-s + (−0.448 − 0.376i)23-s + (0.373 − 2.12i)25-s + (−0.903 + 0.521i)27-s + (0.486 + 1.33i)29-s + (0.615 + 0.355i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.679 + 0.733i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.679 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.16811 - 0.946627i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.16811 - 0.946627i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-360. - 1.73i)T \) |
| good | 3 | \( 1 + (-5.99 + 1.05i)T + (76.1 - 27.7i)T^{2} \) |
| 5 | \( 1 + (-34.0 + 28.5i)T + (108. - 615. i)T^{2} \) |
| 7 | \( 1 + (4.84 - 8.39i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (85.0 + 147. i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-264. - 46.5i)T + (2.68e4 + 9.76e3i)T^{2} \) |
| 17 | \( 1 + (15.8 + 5.78i)T + (6.39e4 + 5.36e4i)T^{2} \) |
| 23 | \( 1 + (237. + 199. i)T + (4.85e4 + 2.75e5i)T^{2} \) |
| 29 | \( 1 + (-408. - 1.12e3i)T + (-5.41e5 + 4.54e5i)T^{2} \) |
| 31 | \( 1 + (-591. - 341. i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 - 75.0iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (1.64e3 - 289. i)T + (2.65e6 - 9.66e5i)T^{2} \) |
| 43 | \( 1 + (1.65e3 - 1.38e3i)T + (5.93e5 - 3.36e6i)T^{2} \) |
| 47 | \( 1 + (3.58e3 - 1.30e3i)T + (3.73e6 - 3.13e6i)T^{2} \) |
| 53 | \( 1 + (-1.51e3 + 1.80e3i)T + (-1.37e6 - 7.77e6i)T^{2} \) |
| 59 | \( 1 + (101. - 279. i)T + (-9.28e6 - 7.78e6i)T^{2} \) |
| 61 | \( 1 + (-4.80e3 - 4.03e3i)T + (2.40e6 + 1.36e7i)T^{2} \) |
| 67 | \( 1 + (-1.48e3 - 4.07e3i)T + (-1.54e7 + 1.29e7i)T^{2} \) |
| 71 | \( 1 + (2.20e3 + 2.63e3i)T + (-4.41e6 + 2.50e7i)T^{2} \) |
| 73 | \( 1 + (32.0 + 181. i)T + (-2.66e7 + 9.71e6i)T^{2} \) |
| 79 | \( 1 + (6.93e3 - 1.22e3i)T + (3.66e7 - 1.33e7i)T^{2} \) |
| 83 | \( 1 + (-1.29e3 + 2.23e3i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (4.59e3 + 810. i)T + (5.89e7 + 2.14e7i)T^{2} \) |
| 97 | \( 1 + (-3.64e3 + 1.00e4i)T + (-6.78e7 - 5.69e7i)T^{2} \) |
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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
−13.59939906790125445359017422539, −13.00959802786865082000065490148, −11.42083593431208016025426543951, −10.05821605977659345543513853790, −8.724882393364008893562373342939, −8.442210629164936475666983765513, −6.15576566081655822010962487697, −5.23759679151692658728930878944, −3.04426901142611593324848081662, −1.33425661596331864226435994263,
2.11683655348629627250646123958, 3.37460550577696662961439147338, 5.61319956149703029836152215121, 6.74317511015758449887960714639, 8.194978599562172053121071187962, 9.668622346581316587094013069370, 10.24851540584975169043637295258, 11.56294629316742052041150857838, 13.42275977099740079465599998944, 13.72673902456873121776002425419
