| L(s) = 1 | + (0.540 − 0.841i)2-s + (1.64 − 0.549i)3-s + (−0.415 − 0.909i)4-s + (0.959 + 0.281i)5-s + (0.426 − 1.67i)6-s + (0.918 + 0.795i)7-s + (−0.989 − 0.142i)8-s + (2.39 − 1.80i)9-s + (0.755 − 0.654i)10-s + (3.79 − 2.44i)11-s + (−1.18 − 1.26i)12-s + (0.541 + 0.624i)13-s + (1.16 − 0.342i)14-s + (1.73 − 0.0639i)15-s + (−0.654 + 0.755i)16-s + (−2.35 + 5.15i)17-s + ⋯ |
| L(s) = 1 | + (0.382 − 0.594i)2-s + (0.948 − 0.316i)3-s + (−0.207 − 0.454i)4-s + (0.429 + 0.125i)5-s + (0.174 − 0.685i)6-s + (0.347 + 0.300i)7-s + (−0.349 − 0.0503i)8-s + (0.799 − 0.601i)9-s + (0.238 − 0.207i)10-s + (1.14 − 0.735i)11-s + (−0.341 − 0.365i)12-s + (0.150 + 0.173i)13-s + (0.311 − 0.0915i)14-s + (0.446 − 0.0165i)15-s + (−0.163 + 0.188i)16-s + (−0.570 + 1.24i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.406 + 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.406 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.33012 - 1.51355i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.33012 - 1.51355i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.540 + 0.841i)T \) |
| 3 | \( 1 + (-1.64 + 0.549i)T \) |
| 5 | \( 1 + (-0.959 - 0.281i)T \) |
| 23 | \( 1 + (4.39 + 1.90i)T \) |
| good | 7 | \( 1 + (-0.918 - 0.795i)T + (0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (-3.79 + 2.44i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-0.541 - 0.624i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (2.35 - 5.15i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (5.34 - 2.44i)T + (12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-6.42 - 2.93i)T + (18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.809 + 5.63i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (2.68 + 9.15i)T + (-31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (-0.674 + 2.29i)T + (-34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (1.09 - 0.157i)T + (41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 - 10.1iT - 47T^{2} \) |
| 53 | \( 1 + (4.13 - 4.77i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (10.1 - 8.83i)T + (8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (11.5 + 1.65i)T + (58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (-3.39 + 5.28i)T + (-27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (-5.79 + 9.01i)T + (-29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-3.89 - 8.53i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (8.25 - 7.15i)T + (11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-4.95 + 1.45i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-2.66 - 18.5i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (2.65 - 9.04i)T + (-81.6 - 52.4i)T^{2} \) |
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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
−10.42380505111241735061965279991, −9.315188220944247414814058861926, −8.726958098620507936366786407281, −7.982004646426216871991260697356, −6.44738548230590281793687243459, −6.08422855844500895544609612724, −4.36133349190803981106274651484, −3.70459901392023171478620083025, −2.37723221242724405192076393264, −1.51593350934396080220640340078,
1.81875786846914480955964752594, 3.12008308494740812307597497432, 4.41467725796613366241980117439, 4.80152031542437892338915079704, 6.41966138545323843612797869330, 7.01843414175893431420194191728, 8.104799565683633978120639641903, 8.817281122202280236427579986176, 9.587826722181215417393521897058, 10.36289605526174705108275282808
