| L(s) = 1 | + (−2.15 + 0.577i)2-s + (0.879 − 1.49i)3-s + (2.57 − 1.48i)4-s + (0.469 + 0.611i)5-s + (−1.03 + 3.72i)6-s + (0.376 − 2.61i)7-s + (−1.53 + 1.53i)8-s + (−1.45 − 2.62i)9-s + (−1.36 − 1.04i)10-s + (0.745 − 0.971i)11-s + (0.0466 − 5.14i)12-s − 0.918·13-s + (0.700 + 5.85i)14-s + (1.32 − 0.162i)15-s + (−0.558 + 0.966i)16-s + (4.12 − 0.0634i)17-s + ⋯ |
| L(s) = 1 | + (−1.52 + 0.408i)2-s + (0.507 − 0.861i)3-s + (1.28 − 0.742i)4-s + (0.209 + 0.273i)5-s + (−0.421 + 1.51i)6-s + (0.142 − 0.989i)7-s + (−0.541 + 0.541i)8-s + (−0.484 − 0.874i)9-s + (−0.430 − 0.330i)10-s + (0.224 − 0.293i)11-s + (0.0134 − 1.48i)12-s − 0.254·13-s + (0.187 + 1.56i)14-s + (0.342 − 0.0418i)15-s + (−0.139 + 0.241i)16-s + (0.999 − 0.0153i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0451 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0451 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.508973 - 0.486495i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.508973 - 0.486495i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.879 + 1.49i)T \) |
| 7 | \( 1 + (-0.376 + 2.61i)T \) |
| 17 | \( 1 + (-4.12 + 0.0634i)T \) |
| good | 2 | \( 1 + (2.15 - 0.577i)T + (1.73 - i)T^{2} \) |
| 5 | \( 1 + (-0.469 - 0.611i)T + (-1.29 + 4.82i)T^{2} \) |
| 11 | \( 1 + (-0.745 + 0.971i)T + (-2.84 - 10.6i)T^{2} \) |
| 13 | \( 1 + 0.918T + 13T^{2} \) |
| 19 | \( 1 + (2.96 - 0.795i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (4.99 + 0.657i)T + (22.2 + 5.95i)T^{2} \) |
| 29 | \( 1 + (-0.788 - 0.326i)T + (20.5 + 20.5i)T^{2} \) |
| 31 | \( 1 + (1.56 - 0.206i)T + (29.9 - 8.02i)T^{2} \) |
| 37 | \( 1 + (-6.94 + 5.32i)T + (9.57 - 35.7i)T^{2} \) |
| 41 | \( 1 + (3.83 + 9.25i)T + (-28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (-0.699 - 0.699i)T + 43iT^{2} \) |
| 47 | \( 1 + (-7.53 - 4.35i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.572 + 2.13i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-0.0323 + 0.120i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-1.42 + 10.8i)T + (-58.9 - 15.7i)T^{2} \) |
| 67 | \( 1 + (1.45 + 2.51i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3.39 - 8.19i)T + (-50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (-0.172 - 1.30i)T + (-70.5 + 18.8i)T^{2} \) |
| 79 | \( 1 + (0.899 - 6.83i)T + (-76.3 - 20.4i)T^{2} \) |
| 83 | \( 1 + (4.23 - 4.23i)T - 83iT^{2} \) |
| 89 | \( 1 + (-12.1 - 7.00i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-16.2 - 6.71i)T + (68.5 + 68.5i)T^{2} \) |
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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
−10.83672391193893020440476922919, −10.13486047330624698967707346916, −9.249401396391151898712455024950, −8.239846982688456863843467475742, −7.67336395755715725394894491048, −6.85080773280392889041327214469, −6.01696177026214987968317229972, −3.87831588993813079648948220355, −2.15403941832628143942866019362, −0.75803216711244339119969791942,
1.84287532478664210269107558125, 2.99174042780858033164659967217, 4.65622499182104979804045454740, 5.86258983691009522143014921644, 7.52454973174985172693125324325, 8.359494659406048753296737252390, 9.035054607634755367139420123260, 9.741368225889963732784635706351, 10.34246962677043626475567793721, 11.43201195336406332691598172771
