| L(s) = 1 | + (1.28 + 0.587i)2-s + (0.859 − 2.92i)3-s + (1.30 + 1.51i)4-s + (2.95 + 4.03i)5-s + (2.82 − 3.26i)6-s + (−1.96 + 13.6i)7-s + (0.796 + 2.71i)8-s + (−0.264 − 0.169i)9-s + (1.42 + 6.92i)10-s + (6.64 − 3.03i)11-s + (5.55 − 2.53i)12-s + (−21.6 + 3.11i)13-s + (−10.5 + 16.4i)14-s + (14.3 − 5.17i)15-s + (−0.569 + 3.95i)16-s + (17.9 − 20.7i)17-s + ⋯ |
| L(s) = 1 | + (0.643 + 0.293i)2-s + (0.286 − 0.976i)3-s + (0.327 + 0.377i)4-s + (0.590 + 0.807i)5-s + (0.471 − 0.543i)6-s + (−0.280 + 1.95i)7-s + (0.0996 + 0.339i)8-s + (−0.0293 − 0.0188i)9-s + (0.142 + 0.692i)10-s + (0.604 − 0.276i)11-s + (0.462 − 0.211i)12-s + (−1.66 + 0.239i)13-s + (−0.754 + 1.17i)14-s + (0.957 − 0.344i)15-s + (−0.0355 + 0.247i)16-s + (1.05 − 1.21i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 - 0.673i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.739 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.43237 + 0.941961i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.43237 + 0.941961i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.28 - 0.587i)T \) |
| 5 | \( 1 + (-2.95 - 4.03i)T \) |
| 23 | \( 1 + (9.09 + 21.1i)T \) |
| good | 3 | \( 1 + (-0.859 + 2.92i)T + (-7.57 - 4.86i)T^{2} \) |
| 7 | \( 1 + (1.96 - 13.6i)T + (-47.0 - 13.8i)T^{2} \) |
| 11 | \( 1 + (-6.64 + 3.03i)T + (79.2 - 91.4i)T^{2} \) |
| 13 | \( 1 + (21.6 - 3.11i)T + (162. - 47.6i)T^{2} \) |
| 17 | \( 1 + (-17.9 + 20.7i)T + (-41.1 - 286. i)T^{2} \) |
| 19 | \( 1 + (-3.89 + 3.37i)T + (51.3 - 357. i)T^{2} \) |
| 29 | \( 1 + (-18.1 + 20.9i)T + (-119. - 832. i)T^{2} \) |
| 31 | \( 1 + (-14.0 + 4.12i)T + (808. - 519. i)T^{2} \) |
| 37 | \( 1 + (25.9 + 16.6i)T + (568. + 1.24e3i)T^{2} \) |
| 41 | \( 1 + (-17.5 + 11.2i)T + (698. - 1.52e3i)T^{2} \) |
| 43 | \( 1 + (-5.99 - 1.76i)T + (1.55e3 + 9.99e2i)T^{2} \) |
| 47 | \( 1 - 47.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-7.47 + 51.9i)T + (-2.69e3 - 791. i)T^{2} \) |
| 59 | \( 1 + (-5.58 - 38.8i)T + (-3.33e3 + 980. i)T^{2} \) |
| 61 | \( 1 + (0.144 + 0.491i)T + (-3.13e3 + 2.01e3i)T^{2} \) |
| 67 | \( 1 + (22.6 - 49.6i)T + (-2.93e3 - 3.39e3i)T^{2} \) |
| 71 | \( 1 + (16.7 - 36.6i)T + (-3.30e3 - 3.80e3i)T^{2} \) |
| 73 | \( 1 + (-49.7 + 43.1i)T + (758. - 5.27e3i)T^{2} \) |
| 79 | \( 1 + (-112. + 16.1i)T + (5.98e3 - 1.75e3i)T^{2} \) |
| 83 | \( 1 + (62.5 + 40.2i)T + (2.86e3 + 6.26e3i)T^{2} \) |
| 89 | \( 1 + (-9.74 + 33.1i)T + (-6.66e3 - 4.28e3i)T^{2} \) |
| 97 | \( 1 + (-50.5 + 32.4i)T + (3.90e3 - 8.55e3i)T^{2} \) |
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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
−12.07397777977800184225860127240, −11.81669308077232770349489433912, −10.00726264649210164629202034990, −9.123928992990786194588206823881, −7.82686529706639020329282031743, −6.88226966554075729339451400369, −6.06687568422494244150372992499, −5.04510222539567092255459684922, −2.80619576350735911637436633846, −2.29028688574142553513111308708,
1.29467728049649548565802826541, 3.48142147283540769466679284112, 4.30070900535950147624761194036, 5.16532965203624047564465445106, 6.70172040737874375124542223584, 7.81484542269624811165197610336, 9.461021390454341221601219978249, 10.07562035473799102350114003914, 10.48442864387231087836383538065, 12.13099852053164825465469810140
