| L(s) = 1 | + (0.386 + 0.348i)2-s + (0.460 + 1.03i)3-s + (−0.180 − 1.72i)4-s + (0.678 + 3.19i)5-s + (−0.182 + 0.560i)6-s + (−1.97 − 1.76i)7-s + (1.14 − 1.56i)8-s + (1.14 − 1.27i)9-s + (−0.848 + 1.47i)10-s + (−2.68 + 1.94i)11-s + (1.69 − 0.980i)12-s + (−0.467 − 1.43i)13-s + (−0.147 − 1.36i)14-s + (−2.99 + 2.17i)15-s + (−2.39 + 0.509i)16-s + (−1.68 − 1.86i)17-s + ⋯ |
| L(s) = 1 | + (0.273 + 0.246i)2-s + (0.266 + 0.597i)3-s + (−0.0903 − 0.860i)4-s + (0.303 + 1.42i)5-s + (−0.0743 + 0.228i)6-s + (−0.744 − 0.667i)7-s + (0.403 − 0.554i)8-s + (0.382 − 0.424i)9-s + (−0.268 + 0.464i)10-s + (−0.809 + 0.587i)11-s + (0.490 − 0.282i)12-s + (−0.129 − 0.399i)13-s + (−0.0393 − 0.365i)14-s + (−0.772 + 0.561i)15-s + (−0.599 + 0.127i)16-s + (−0.407 − 0.452i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.539i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.841 - 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.05703 + 0.309718i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05703 + 0.309718i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (1.97 + 1.76i)T \) |
| 11 | \( 1 + (2.68 - 1.94i)T \) |
| good | 2 | \( 1 + (-0.386 - 0.348i)T + (0.209 + 1.98i)T^{2} \) |
| 3 | \( 1 + (-0.460 - 1.03i)T + (-2.00 + 2.22i)T^{2} \) |
| 5 | \( 1 + (-0.678 - 3.19i)T + (-4.56 + 2.03i)T^{2} \) |
| 13 | \( 1 + (0.467 + 1.43i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (1.68 + 1.86i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-0.167 + 1.59i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (1.98 + 3.43i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.95 - 6.82i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (1.98 - 9.33i)T + (-28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (-6.07 - 2.70i)T + (24.7 + 27.4i)T^{2} \) |
| 41 | \( 1 + (-3.25 - 2.36i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 5.21iT - 43T^{2} \) |
| 47 | \( 1 + (-0.345 - 0.0363i)T + (45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (3.34 + 0.711i)T + (48.4 + 21.5i)T^{2} \) |
| 59 | \( 1 + (0.669 - 0.0704i)T + (57.7 - 12.2i)T^{2} \) |
| 61 | \( 1 + (7.31 - 1.55i)T + (55.7 - 24.8i)T^{2} \) |
| 67 | \( 1 + (-4.14 + 7.17i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.797 + 2.45i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (1.24 + 11.7i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (8.76 + 7.88i)T + (8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (4.17 - 12.8i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-6.53 + 3.77i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (4.04 - 1.31i)T + (78.4 - 57.0i)T^{2} \) |
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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
−14.61660389542748683789207180630, −13.88452706532102545145397273871, −12.70855189496569062596972370895, −10.56147009310804922710355137649, −10.42477460086366221958245562046, −9.381839344551635392215120942992, −7.14245911322825024376357311865, −6.45129701530329809394080197792, −4.71141616535739840492016687080, −3.05271661548077276755170086318,
2.36254059093596835590928394521, 4.35269928298703442270642148815, 5.85311012685086227722229645340, 7.74799674684334982897426945243, 8.533025757943067018084175536235, 9.682078284532335716428231209265, 11.53774010791457082943701656845, 12.68133506705343717074206588555, 13.02660632863372162187577325532, 13.77896369806382425232884054963
