| L(s) = 1 | + (0.967 − 2.65i)2-s + (−7.41 + 1.30i)3-s + (−6.12 − 5.14i)4-s + (−15.9 + 13.4i)5-s + (−3.69 + 20.9i)6-s + (−18.4 + 31.8i)7-s + (−19.5 + 11.3i)8-s + (−22.8 + 8.33i)9-s + (20.1 + 55.4i)10-s + (−28.3 − 49.1i)11-s + (52.1 + 30.1i)12-s + (−38.4 − 6.77i)13-s + (66.9 + 79.7i)14-s + (100. − 120. i)15-s + (11.1 + 63.0i)16-s + (−35.2 − 12.8i)17-s + ⋯ |
| L(s) = 1 | + (0.241 − 0.664i)2-s + (−0.823 + 0.145i)3-s + (−0.383 − 0.321i)4-s + (−0.638 + 0.536i)5-s + (−0.102 + 0.582i)6-s + (−0.375 + 0.650i)7-s + (−0.306 + 0.176i)8-s + (−0.282 + 0.102i)9-s + (0.201 + 0.554i)10-s + (−0.234 − 0.406i)11-s + (0.362 + 0.209i)12-s + (−0.227 − 0.0400i)13-s + (0.341 + 0.406i)14-s + (0.448 − 0.534i)15-s + (0.0434 + 0.246i)16-s + (−0.122 − 0.0444i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.676 - 0.736i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.676 - 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.0755706 + 0.172109i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0755706 + 0.172109i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.967 + 2.65i)T \) |
| 19 | \( 1 + (120. + 340. i)T \) |
| good | 3 | \( 1 + (7.41 - 1.30i)T + (76.1 - 27.7i)T^{2} \) |
| 5 | \( 1 + (15.9 - 13.4i)T + (108. - 615. i)T^{2} \) |
| 7 | \( 1 + (18.4 - 31.8i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (28.3 + 49.1i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (38.4 + 6.77i)T + (2.68e4 + 9.76e3i)T^{2} \) |
| 17 | \( 1 + (35.2 + 12.8i)T + (6.39e4 + 5.36e4i)T^{2} \) |
| 23 | \( 1 + (273. + 229. i)T + (4.85e4 + 2.75e5i)T^{2} \) |
| 29 | \( 1 + (-134. - 370. i)T + (-5.41e5 + 4.54e5i)T^{2} \) |
| 31 | \( 1 + (-411. - 237. i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 - 2.40e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (3.21e3 - 566. i)T + (2.65e6 - 9.66e5i)T^{2} \) |
| 43 | \( 1 + (1.00e3 - 844. i)T + (5.93e5 - 3.36e6i)T^{2} \) |
| 47 | \( 1 + (-331. + 120. i)T + (3.73e6 - 3.13e6i)T^{2} \) |
| 53 | \( 1 + (-1.17e3 + 1.39e3i)T + (-1.37e6 - 7.77e6i)T^{2} \) |
| 59 | \( 1 + (-1.40e3 + 3.86e3i)T + (-9.28e6 - 7.78e6i)T^{2} \) |
| 61 | \( 1 + (-333. - 279. i)T + (2.40e6 + 1.36e7i)T^{2} \) |
| 67 | \( 1 + (-1.03e3 - 2.83e3i)T + (-1.54e7 + 1.29e7i)T^{2} \) |
| 71 | \( 1 + (2.89e3 + 3.45e3i)T + (-4.41e6 + 2.50e7i)T^{2} \) |
| 73 | \( 1 + (740. + 4.19e3i)T + (-2.66e7 + 9.71e6i)T^{2} \) |
| 79 | \( 1 + (1.08e4 - 1.90e3i)T + (3.66e7 - 1.33e7i)T^{2} \) |
| 83 | \( 1 + (-6.06e3 + 1.04e4i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (9.49e3 + 1.67e3i)T + (5.89e7 + 2.14e7i)T^{2} \) |
| 97 | \( 1 + (5.37e3 - 1.47e4i)T + (-6.78e7 - 5.69e7i)T^{2} \) |
| 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
−15.89630062116666266157152593137, −14.86971253481162635976184711722, −13.39747973315938244094509052681, −12.01026698909344436551904320614, −11.32631295946640267416756248818, −10.21860342642952902447004472432, −8.523213064192498217833649633525, −6.47978077773814632000437037422, −4.98999342825209535729182608975, −2.98868998975068100483887782809,
0.12590889468043474727551696291, 4.10097540697215866336398615192, 5.64788061733591131565455990452, 7.06125671309328315562638393765, 8.409815000679583628623638900325, 10.17899295630167737857498416917, 11.77553305868027669730435809206, 12.62167691664932778312470571631, 13.96067135816965437045586178523, 15.33774510762093546489444962563
