| L(s) = 1 | + (−0.766 + 0.642i)2-s + (0.0916 + 0.0333i)3-s + (0.173 − 0.984i)4-s + (0.327 + 1.85i)5-s + (−0.0916 + 0.0333i)6-s + (1.90 − 3.30i)7-s + (0.500 + 0.866i)8-s + (−2.29 − 1.92i)9-s + (−1.44 − 1.21i)10-s + (−0.5 − 0.866i)11-s + (0.0487 − 0.0845i)12-s + (5.98 − 2.18i)13-s + (0.661 + 3.75i)14-s + (−0.0319 + 0.181i)15-s + (−0.939 − 0.342i)16-s + (1.94 − 1.63i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 + 0.454i)2-s + (0.0529 + 0.0192i)3-s + (0.0868 − 0.492i)4-s + (0.146 + 0.831i)5-s + (−0.0374 + 0.0136i)6-s + (0.720 − 1.24i)7-s + (0.176 + 0.306i)8-s + (−0.763 − 0.640i)9-s + (−0.457 − 0.383i)10-s + (−0.150 − 0.261i)11-s + (0.0140 − 0.0243i)12-s + (1.66 − 0.604i)13-s + (0.176 + 1.00i)14-s + (−0.00826 + 0.0468i)15-s + (−0.234 − 0.0855i)16-s + (0.471 − 0.395i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.15958 - 0.0577283i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15958 - 0.0577283i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.766 - 0.642i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (4.20 - 1.14i)T \) |
| good | 3 | \( 1 + (-0.0916 - 0.0333i)T + (2.29 + 1.92i)T^{2} \) |
| 5 | \( 1 + (-0.327 - 1.85i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (-1.90 + 3.30i)T + (-3.5 - 6.06i)T^{2} \) |
| 13 | \( 1 + (-5.98 + 2.18i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.94 + 1.63i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (0.793 - 4.49i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-1.50 - 1.26i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-5.05 + 8.74i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 6.50T + 37T^{2} \) |
| 41 | \( 1 + (-1.11 - 0.405i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.68 - 9.56i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-0.109 - 0.0922i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-0.608 + 3.44i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (7.12 - 5.98i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (1.16 - 6.58i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (7.20 + 6.04i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (1.64 + 9.34i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (3.40 + 1.23i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-0.796 - 0.290i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-1.00 + 1.74i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.33 + 0.849i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (11.3 - 9.50i)T + (16.8 - 95.5i)T^{2} \) |
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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
−10.93904026071523531687468874600, −10.45073507329069782301066818643, −9.344789812231248334469467234871, −8.202857022617500719130536559430, −7.69546790797155709245686762895, −6.43351309125428498206850606058, −5.87411485660412861492124767079, −4.22575543512298191529180071696, −3.03664576348025423345114373265, −1.03675401258045268672041952739,
1.53368407746826528679345018602, 2.66933257902152321178550744839, 4.36507398097778228944826380346, 5.41209733099023443583204852407, 6.41579252555590032849605912022, 8.145046663207845274202103172617, 8.594149581213060437011417683906, 9.028295370046791054574867809025, 10.48795906701903810390070710289, 11.15464390044155530485166975914
