| 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
−13.72673902456873121776002425419, −13.42275977099740079465599998944, −11.56294629316742052041150857838, −10.24851540584975169043637295258, −9.668622346581316587094013069370, −8.194978599562172053121071187962, −6.74317511015758449887960714639, −5.61319956149703029836152215121, −3.37460550577696662961439147338, −2.11683655348629627250646123958,
1.33425661596331864226435994263, 3.04426901142611593324848081662, 5.23759679151692658728930878944, 6.15576566081655822010962487697, 8.442210629164936475666983765513, 8.724882393364008893562373342939, 10.05821605977659345543513853790, 11.42083593431208016025426543951, 13.00959802786865082000065490148, 13.59939906790125445359017422539
