| L(s) = 1 | + (−0.311 + 1.46i)2-s + (0.599 + 0.0630i)3-s + (−0.220 − 0.0983i)4-s + (1.75 − 1.58i)5-s + (−0.279 + 0.858i)6-s + (−2.26 + 1.37i)7-s + (−1.54 + 2.12i)8-s + (−2.57 − 0.548i)9-s + (1.76 + 3.06i)10-s + (0.972 − 3.17i)11-s + (−0.126 − 0.0729i)12-s + (−0.304 − 0.938i)13-s + (−1.31 − 3.73i)14-s + (1.15 − 0.837i)15-s + (−2.96 − 3.28i)16-s + (7.13 − 1.51i)17-s + ⋯ |
| L(s) = 1 | + (−0.220 + 1.03i)2-s + (0.346 + 0.0364i)3-s + (−0.110 − 0.0491i)4-s + (0.785 − 0.707i)5-s + (−0.113 + 0.350i)6-s + (−0.854 + 0.519i)7-s + (−0.547 + 0.752i)8-s + (−0.859 − 0.182i)9-s + (0.559 + 0.969i)10-s + (0.293 − 0.956i)11-s + (−0.0364 − 0.0210i)12-s + (−0.0845 − 0.260i)13-s + (−0.350 − 0.999i)14-s + (0.297 − 0.216i)15-s + (−0.740 − 0.822i)16-s + (1.72 − 0.367i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.370 - 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.370 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.813743 + 0.551692i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.813743 + 0.551692i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.26 - 1.37i)T \) |
| 11 | \( 1 + (-0.972 + 3.17i)T \) |
| good | 2 | \( 1 + (0.311 - 1.46i)T + (-1.82 - 0.813i)T^{2} \) |
| 3 | \( 1 + (-0.599 - 0.0630i)T + (2.93 + 0.623i)T^{2} \) |
| 5 | \( 1 + (-1.75 + 1.58i)T + (0.522 - 4.97i)T^{2} \) |
| 13 | \( 1 + (0.304 + 0.938i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-7.13 + 1.51i)T + (15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (4.47 - 1.99i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (0.0581 - 0.100i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.73 + 6.51i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-2.70 - 2.43i)T + (3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (-0.386 - 3.67i)T + (-36.1 + 7.69i)T^{2} \) |
| 41 | \( 1 + (-3.09 - 2.24i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 5.00iT - 43T^{2} \) |
| 47 | \( 1 + (-3.27 - 7.34i)T + (-31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (2.58 - 2.87i)T + (-5.54 - 52.7i)T^{2} \) |
| 59 | \( 1 + (-0.577 + 1.29i)T + (-39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (0.158 + 0.176i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (2.19 + 3.80i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (2.39 - 7.35i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (1.47 + 0.656i)T + (48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (2.53 - 11.9i)T + (-72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (-1.29 + 3.97i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (11.4 + 6.59i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-16.5 + 5.39i)T + (78.4 - 57.0i)T^{2} \) |
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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
−14.74412724073430664895248350212, −13.92516667152992069662141239621, −12.69242380150798420218024681870, −11.56693401363865199907425435946, −9.730514507175321198941975693746, −8.846692814698370594870494948609, −7.941343268240910751662307645403, −6.11351567786514078470378564520, −5.67406139267155136199104280900, −2.98056261761740766793844189187,
2.24180556930121243633068016491, 3.53241990166745996999319025785, 6.02552652783389428834555235806, 7.17944044155128078665168227813, 9.118724856447721295384453680708, 10.05984943897140488089653628905, 10.70997478115046012219991686014, 12.06796604383449095304492259874, 13.02939546618785438234488785384, 14.23250315378314385756310933721
