| L(s) = 1 | + (−0.387 − 2.69i)2-s + (0.415 − 0.909i)3-s + (−5.20 + 1.52i)4-s + (0.654 − 0.755i)5-s + (−2.61 − 0.767i)6-s + (−4.11 + 2.64i)7-s + (3.88 + 8.49i)8-s + (−0.654 − 0.755i)9-s + (−2.29 − 1.47i)10-s + (−0.268 + 1.87i)11-s + (−0.772 + 5.37i)12-s + (−2.01 − 1.29i)13-s + (8.73 + 10.0i)14-s + (−0.415 − 0.909i)15-s + (12.2 − 7.89i)16-s + (−6.87 − 2.01i)17-s + ⋯ |
| L(s) = 1 | + (−0.274 − 1.90i)2-s + (0.239 − 0.525i)3-s + (−2.60 + 0.764i)4-s + (0.292 − 0.337i)5-s + (−1.06 − 0.313i)6-s + (−1.55 + 1.00i)7-s + (1.37 + 3.00i)8-s + (−0.218 − 0.251i)9-s + (−0.725 − 0.465i)10-s + (−0.0810 + 0.563i)11-s + (−0.223 + 1.55i)12-s + (−0.559 − 0.359i)13-s + (2.33 + 2.69i)14-s + (−0.107 − 0.234i)15-s + (3.07 − 1.97i)16-s + (−1.66 − 0.489i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.476 - 0.879i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.476 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.162571 + 0.0968031i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.162571 + 0.0968031i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.415 + 0.909i)T \) |
| 5 | \( 1 + (-0.654 + 0.755i)T \) |
| 23 | \( 1 + (-2.12 + 4.29i)T \) |
| good | 2 | \( 1 + (0.387 + 2.69i)T + (-1.91 + 0.563i)T^{2} \) |
| 7 | \( 1 + (4.11 - 2.64i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (0.268 - 1.87i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (2.01 + 1.29i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (6.87 + 2.01i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (1.06 - 0.313i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (2.63 + 0.773i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (0.599 + 1.31i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (5.61 + 6.48i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (-0.290 + 0.335i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (0.198 - 0.435i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 - 7.64T + 47T^{2} \) |
| 53 | \( 1 + (2.82 - 1.81i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (-6.96 - 4.47i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (-1.03 - 2.26i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (0.109 + 0.764i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (-1.27 - 8.84i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (7.49 - 2.20i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (4.63 + 2.97i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (7.09 + 8.18i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (6.61 - 14.4i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (1.34 - 1.55i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.73660417021433393434053406521, −9.800844337399280139420202936048, −9.127096088848718302205364117983, −8.617749865811630816040278470032, −6.99460788119332399996817483556, −5.52989257711154673176248628145, −4.19531234816052103198727338099, −2.77349932022880866002046519823, −2.21931799734082634480432807772, −0.13303444299762353656736369032,
3.50491643171080357331696677734, 4.52301135432545771782607813091, 5.81446025803667965608093895312, 6.72830300069727725315817683269, 7.19583515499434021100077843944, 8.536485661109808513247137457147, 9.322036082711628870907708163398, 9.972815163554905098320421240397, 10.84520275015398238032150593812, 12.91895974054097262572635851827
