L(s) = 1 | + (−3.24 + 4.05i)3-s + (3.20 + 5.55i)5-s + (−14.8 + 11.1i)7-s + (−5.89 − 26.3i)9-s + (−16.2 − 9.39i)11-s − 3.82i·13-s + (−32.9 − 5.03i)15-s + (−8.12 + 14.0i)17-s + (−129. + 74.9i)19-s + (3.01 − 96.1i)21-s + (109. − 63.3i)23-s + (41.9 − 72.6i)25-s + (126. + 61.6i)27-s − 168. i·29-s + (43.4 + 25.0i)31-s + ⋯ |
L(s) = 1 | + (−0.625 + 0.780i)3-s + (0.286 + 0.496i)5-s + (−0.799 + 0.600i)7-s + (−0.218 − 0.975i)9-s + (−0.446 − 0.257i)11-s − 0.0816i·13-s + (−0.567 − 0.0867i)15-s + (−0.115 + 0.200i)17-s + (−1.56 + 0.904i)19-s + (0.0313 − 0.999i)21-s + (0.994 − 0.573i)23-s + (0.335 − 0.581i)25-s + (0.898 + 0.439i)27-s − 1.07i·29-s + (0.251 + 0.145i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0952 + 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0952 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.3373914868\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3373914868\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (3.24 - 4.05i)T \) |
| 7 | \( 1 + (14.8 - 11.1i)T \) |
good | 5 | \( 1 + (-3.20 - 5.55i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (16.2 + 9.39i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 3.82iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (8.12 - 14.0i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (129. - 74.9i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-109. + 63.3i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 168. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (-43.4 - 25.0i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-24.7 - 42.8i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 19.9T + 6.89e4T^{2} \) |
| 43 | \( 1 + 5.14T + 7.95e4T^{2} \) |
| 47 | \( 1 + (308. + 534. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-242. - 140. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-274. + 475. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (507. - 293. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-378. + 655. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 351. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (530. + 306. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (505. + 876. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 1.01e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-178. - 309. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 - 228. iT - 9.12e5T^{2} \) |
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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
−10.58369470719917559619660463041, −10.28317011752287469544318781950, −9.186990216220456191738037803258, −8.288526375766430971702824060586, −6.56765433542586443269251285640, −6.13877700820626168601299957850, −4.96389900504380557239522454002, −3.69107057820647225063730807252, −2.49986552914975391124217365810, −0.13868954231622725648851092688,
1.22328633290270394982432762442, 2.75320744227479287227999478792, 4.49663205027693978435847478337, 5.49885803281426991155879322043, 6.66511282023917731036665299007, 7.23403912254538820515880065793, 8.508168284799488446612173712513, 9.488452471157976213166762113523, 10.62798575942207389798989977025, 11.24044227779655621388994145350