| L(s) = 1 | + (1.53 + 4.70i)2-s + (−13.3 + 9.71i)4-s + (9.83 − 5.31i)5-s + 28.2·7-s + (−34.1 − 24.8i)8-s + (40.0 + 38.1i)10-s + (3.06 + 9.43i)11-s + (−28.5 + 87.7i)13-s + (43.2 + 133. i)14-s + (23.7 − 73.0i)16-s + (−15.8 − 11.5i)17-s + (62.2 + 45.2i)19-s + (−79.8 + 166. i)20-s + (−39.7 + 28.8i)22-s + (−31.3 − 96.3i)23-s + ⋯ |
| L(s) = 1 | + (0.541 + 1.66i)2-s + (−1.67 + 1.21i)4-s + (0.879 − 0.475i)5-s + 1.52·7-s + (−1.50 − 1.09i)8-s + (1.26 + 1.20i)10-s + (0.0840 + 0.258i)11-s + (−0.608 + 1.87i)13-s + (0.826 + 2.54i)14-s + (0.370 − 1.14i)16-s + (−0.226 − 0.164i)17-s + (0.751 + 0.545i)19-s + (−0.893 + 1.86i)20-s + (−0.385 + 0.279i)22-s + (−0.283 − 0.873i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.869 - 0.494i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.869 - 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.710233 + 2.68413i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.710233 + 2.68413i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-9.83 + 5.31i)T \) |
| good | 2 | \( 1 + (-1.53 - 4.70i)T + (-6.47 + 4.70i)T^{2} \) |
| 7 | \( 1 - 28.2T + 343T^{2} \) |
| 11 | \( 1 + (-3.06 - 9.43i)T + (-1.07e3 + 782. i)T^{2} \) |
| 13 | \( 1 + (28.5 - 87.7i)T + (-1.77e3 - 1.29e3i)T^{2} \) |
| 17 | \( 1 + (15.8 + 11.5i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-62.2 - 45.2i)T + (2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 + (31.3 + 96.3i)T + (-9.84e3 + 7.15e3i)T^{2} \) |
| 29 | \( 1 + (-175. + 127. i)T + (7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (84.9 + 61.7i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (102. - 315. i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (29.3 - 90.4i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 + 67.8T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-95.0 + 69.0i)T + (3.20e4 - 9.87e4i)T^{2} \) |
| 53 | \( 1 + (180. - 130. i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (53.9 - 166. i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (145. + 449. i)T + (-1.83e5 + 1.33e5i)T^{2} \) |
| 67 | \( 1 + (-101. - 73.8i)T + (9.29e4 + 2.86e5i)T^{2} \) |
| 71 | \( 1 + (-750. + 545. i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-31.2 - 96.2i)T + (-3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (238. - 173. i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (899. + 653. i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 + (39.4 + 121. i)T + (-5.70e5 + 4.14e5i)T^{2} \) |
| 97 | \( 1 + (-1.23e3 + 894. i)T + (2.82e5 - 8.68e5i)T^{2} \) |
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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
−12.41084269992557823214328185882, −11.53879291299621601907942048760, −9.886905247093919952610950555481, −8.857059658423094681558664595242, −8.088357233321655059933403644930, −7.00702025677936748780008889985, −6.07800467839738812895601902963, −4.83228512132218673097379937942, −4.51129034400176106215768048900, −1.84273104313669424599142081119,
1.09625818191368897525473036083, 2.29099351190007407959033057164, 3.36333001921124426190393435659, 4.99965071327231810894808353197, 5.50797848524927667286063634282, 7.48529139705954074502729668908, 8.795875810234777729238723797785, 9.943548653141982351803352793647, 10.68164710560484737841411753340, 11.24326971573853156668999382995
