| L(s) = 1 | + (−1.53 − 4.70i)2-s + (−2.42 + 1.76i)3-s + (−13.3 + 9.71i)4-s + (−9.83 + 5.31i)5-s + (12.0 + 8.73i)6-s + 28.2·7-s + (34.1 + 24.8i)8-s + (2.78 − 8.55i)9-s + (40.0 + 38.1i)10-s + (−3.06 − 9.43i)11-s + (15.3 − 47.1i)12-s + (−28.5 + 87.7i)13-s + (−43.2 − 133. i)14-s + (14.5 − 30.2i)15-s + (23.7 − 73.0i)16-s + (15.8 + 11.5i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 − 1.66i)2-s + (−0.467 + 0.339i)3-s + (−1.67 + 1.21i)4-s + (−0.879 + 0.475i)5-s + (0.817 + 0.594i)6-s + 1.52·7-s + (1.50 + 1.09i)8-s + (0.103 − 0.317i)9-s + (1.26 + 1.20i)10-s + (−0.0840 − 0.258i)11-s + (0.368 − 1.13i)12-s + (−0.608 + 1.87i)13-s + (−0.826 − 2.54i)14-s + (0.249 − 0.520i)15-s + (0.370 − 1.14i)16-s + (0.226 + 0.164i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.110i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.993 - 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.649635 + 0.0360808i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.649635 + 0.0360808i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.42 - 1.76i)T \) |
| 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
−13.94901996781140994208033273266, −12.24347138415832075524124281164, −11.38429786409651140337987955416, −11.25151995308069488633364398260, −9.882257819581406603996089682916, −8.688119313518725087057787201522, −7.42349992482875930620873034446, −4.82105203579581108079956488296, −3.66411547826428296003908377847, −1.66043630414913995821204244049,
0.57067805206936034152447885873, 4.81649681050252270985237525965, 5.51774255704679017620454968489, 7.50070347210644432711701884915, 7.72218477241647515417234010398, 8.884336551390992966731409912467, 10.57045563231360101264034611874, 11.86227431990177133132039787670, 13.12867569575212541049492556969, 14.62326031804206645314411093870
