| L(s) = 1 | + (0.395 − 2.75i)2-s + (0.0237 + 0.0520i)3-s + (−5.49 − 1.61i)4-s + (−0.654 − 0.755i)5-s + (0.152 − 0.0448i)6-s + (1.15 + 0.743i)7-s + (−4.29 + 9.40i)8-s + (1.96 − 2.26i)9-s + (−2.33 + 1.50i)10-s + (−0.191 − 1.32i)11-s + (−0.0466 − 0.324i)12-s + (3.77 − 2.42i)13-s + (2.50 − 2.88i)14-s + (0.0237 − 0.0520i)15-s + (14.5 + 9.35i)16-s + (−4.59 + 1.34i)17-s + ⋯ |
| L(s) = 1 | + (0.279 − 1.94i)2-s + (0.0137 + 0.0300i)3-s + (−2.74 − 0.806i)4-s + (−0.292 − 0.337i)5-s + (0.0623 − 0.0183i)6-s + (0.437 + 0.281i)7-s + (−1.51 + 3.32i)8-s + (0.654 − 0.754i)9-s + (−0.739 + 0.475i)10-s + (−0.0576 − 0.400i)11-s + (−0.0134 − 0.0936i)12-s + (1.04 − 0.672i)13-s + (0.668 − 0.772i)14-s + (0.00614 − 0.0134i)15-s + (3.63 + 2.33i)16-s + (−1.11 + 0.326i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.232i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.972 + 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.119523 - 1.01607i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.119523 - 1.01607i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.654 + 0.755i)T \) |
| 23 | \( 1 + (1.09 - 4.66i)T \) |
| good | 2 | \( 1 + (-0.395 + 2.75i)T + (-1.91 - 0.563i)T^{2} \) |
| 3 | \( 1 + (-0.0237 - 0.0520i)T + (-1.96 + 2.26i)T^{2} \) |
| 7 | \( 1 + (-1.15 - 0.743i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (0.191 + 1.32i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (-3.77 + 2.42i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (4.59 - 1.34i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-4.28 - 1.25i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-0.946 + 0.277i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-0.902 + 1.97i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (2.99 - 3.45i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (-1.68 - 1.94i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-3.33 - 7.30i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + 3.65T + 47T^{2} \) |
| 53 | \( 1 + (-0.752 - 0.483i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-2.03 + 1.30i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (3.37 - 7.38i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (1.68 - 11.7i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (1.07 - 7.48i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (8.00 + 2.35i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-11.4 + 7.37i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-2.84 + 3.28i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (6.62 + 14.5i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (11.2 + 13.0i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−12.98033743753542017103340698397, −11.85964518528131726513316335513, −11.27653579653869709658967757601, −10.16954780888235513063231381945, −9.153561803548032552703548800530, −8.233222639263093522740574602357, −5.72731418725862317306623629331, −4.35494340750079336008293832437, −3.27948342623133724040115076085, −1.31893951397519543116322122790,
4.08241321859574233209475007783, 4.99634405594196552883025048102, 6.53519358931388591967577843761, 7.29402643895083478496295722220, 8.262009507556154133600180199205, 9.322192308930006599219445156310, 10.79998233372392059700186224888, 12.45435429469124782680848970039, 13.69126499458971966432517874460, 13.98518836547225495659640172150
