| L(s) = 1 | + (0.891 + 0.453i)2-s + (−0.216 + 0.0342i)3-s + (0.587 + 0.809i)4-s + (1.60 − 1.55i)5-s + (−0.208 − 0.0677i)6-s + (−0.0165 + 0.104i)7-s + (0.156 + 0.987i)8-s + (−2.80 + 0.912i)9-s + (2.13 − 0.653i)10-s + (−0.733 + 3.23i)11-s + (−0.154 − 0.154i)12-s + (0.904 − 1.77i)13-s + (−0.0623 + 0.0858i)14-s + (−0.294 + 0.391i)15-s + (−0.309 + 0.951i)16-s + (−2.43 − 4.77i)17-s + ⋯ |
| L(s) = 1 | + (0.630 + 0.321i)2-s + (−0.124 + 0.0197i)3-s + (0.293 + 0.404i)4-s + (0.719 − 0.694i)5-s + (−0.0850 − 0.0276i)6-s + (−0.00627 + 0.0396i)7-s + (0.0553 + 0.349i)8-s + (−0.935 + 0.304i)9-s + (0.676 − 0.206i)10-s + (−0.221 + 0.975i)11-s + (−0.0447 − 0.0447i)12-s + (0.250 − 0.492i)13-s + (−0.0166 + 0.0229i)14-s + (−0.0761 + 0.101i)15-s + (−0.0772 + 0.237i)16-s + (−0.590 − 1.15i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.38468 + 0.233969i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.38468 + 0.233969i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.891 - 0.453i)T \) |
| 5 | \( 1 + (-1.60 + 1.55i)T \) |
| 11 | \( 1 + (0.733 - 3.23i)T \) |
| good | 3 | \( 1 + (0.216 - 0.0342i)T + (2.85 - 0.927i)T^{2} \) |
| 7 | \( 1 + (0.0165 - 0.104i)T + (-6.65 - 2.16i)T^{2} \) |
| 13 | \( 1 + (-0.904 + 1.77i)T + (-7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (2.43 + 4.77i)T + (-9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (3.72 + 2.70i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (1.26 - 1.26i)T - 23iT^{2} \) |
| 29 | \( 1 + (3.14 - 2.28i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-2.80 - 8.62i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-9.02 - 1.42i)T + (35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (-0.361 + 0.498i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (-2.10 - 2.10i)T + 43iT^{2} \) |
| 47 | \( 1 + (1.38 + 8.76i)T + (-44.6 + 14.5i)T^{2} \) |
| 53 | \( 1 + (7.17 + 3.65i)T + (31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (-6.35 - 8.75i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (0.692 + 0.225i)T + (49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-1.43 - 1.43i)T + 67iT^{2} \) |
| 71 | \( 1 + (-0.219 + 0.674i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-13.1 - 2.08i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (2.31 + 7.12i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (5.07 - 2.58i)T + (48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + 0.464iT - 89T^{2} \) |
| 97 | \( 1 + (-0.809 + 1.58i)T + (-57.0 - 78.4i)T^{2} \) |
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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
−13.62542200034944968053128758596, −12.90600254898014341719712479939, −11.87390418019306810548129570525, −10.69256277768221409333934122891, −9.337308997186611246391727342555, −8.273445928115001437062856847957, −6.81668911293992165081744974438, −5.53142163788086213289472564988, −4.66049585583697452439221413148, −2.52631805048186429456721447383,
2.40922113549905990005983458815, 3.92290814975182460326133754362, 5.90619945472720729684906026370, 6.28473410048465442218126267251, 8.199014162234236571317396086381, 9.532480840050715073712427578109, 10.84824128737709899598174220021, 11.30856090100904267670878059362, 12.73431595524869721327896897811, 13.66041877368470216230329732452
