| L(s) = 1 | + (−0.173 + 0.984i)2-s + (2.43 + 2.04i)3-s + (−0.939 − 0.342i)4-s + (3.24 − 1.18i)5-s + (−2.43 + 2.04i)6-s + (−1.75 − 3.03i)7-s + (0.5 − 0.866i)8-s + (1.23 + 7.00i)9-s + (0.599 + 3.39i)10-s + (0.5 − 0.866i)11-s + (−1.59 − 2.75i)12-s + (−1.33 + 1.12i)13-s + (3.29 − 1.19i)14-s + (10.3 + 3.75i)15-s + (0.766 + 0.642i)16-s + (−0.714 + 4.05i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 + 0.696i)2-s + (1.40 + 1.18i)3-s + (−0.469 − 0.171i)4-s + (1.45 − 0.528i)5-s + (−0.994 + 0.834i)6-s + (−0.662 − 1.14i)7-s + (0.176 − 0.306i)8-s + (0.411 + 2.33i)9-s + (0.189 + 1.07i)10-s + (0.150 − 0.261i)11-s + (−0.459 − 0.795i)12-s + (−0.371 + 0.311i)13-s + (0.880 − 0.320i)14-s + (2.66 + 0.969i)15-s + (0.191 + 0.160i)16-s + (−0.173 + 0.982i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.135 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.135 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.65818 + 1.44698i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.65818 + 1.44698i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.173 - 0.984i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (4.00 - 1.71i)T \) |
| good | 3 | \( 1 + (-2.43 - 2.04i)T + (0.520 + 2.95i)T^{2} \) |
| 5 | \( 1 + (-3.24 + 1.18i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (1.75 + 3.03i)T + (-3.5 + 6.06i)T^{2} \) |
| 13 | \( 1 + (1.33 - 1.12i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.714 - 4.05i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-5.27 - 1.92i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (1.46 + 8.32i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (3.12 + 5.42i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 + (-5.02 - 4.21i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (0.0123 - 0.00449i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (1.53 + 8.68i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (3.52 + 1.28i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (0.620 - 3.51i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-7.32 - 2.66i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (2.00 + 11.3i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (1.24 - 0.453i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-5.97 - 5.01i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-1.77 - 1.48i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (3.90 + 6.75i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (3.99 - 3.34i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (1.31 - 7.46i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−10.76388742850985212461460995048, −10.06337657713490724156058193025, −9.540870657751235618769249786097, −8.868903626962670606775321522520, −7.991209725121703068856649794403, −6.77079138649379445410740473405, −5.62362171115024087737799197687, −4.44706681323741094222767574580, −3.63071733661630780797178715716, −2.03170232208698580281292032147,
1.72194708010422588694380852819, 2.60587968127782605742354666591, 3.11383144834716616486875373193, 5.30669807121096465287988775208, 6.58444243059262981401077334057, 7.16653605141995642391612925971, 8.775619989404851697203272211450, 9.028333846319531428918174235713, 9.791650430901851870203582119113, 10.90179883708828712129001397340
