| L(s) = 1 | + (1.65 − 1.06i)2-s + (−0.142 + 0.989i)3-s + (0.770 − 1.68i)4-s + (0.959 − 0.281i)5-s + (0.815 + 1.78i)6-s + (3.04 + 3.51i)7-s + (0.0408 + 0.283i)8-s + (−0.959 − 0.281i)9-s + (1.28 − 1.48i)10-s + (−4.96 − 3.18i)11-s + (1.55 + 1.00i)12-s + (0.663 − 0.765i)13-s + (8.76 + 2.57i)14-s + (0.142 + 0.989i)15-s + (2.79 + 3.22i)16-s + (0.628 + 1.37i)17-s + ⋯ |
| L(s) = 1 | + (1.16 − 0.750i)2-s + (−0.0821 + 0.571i)3-s + (0.385 − 0.843i)4-s + (0.429 − 0.125i)5-s + (0.332 + 0.729i)6-s + (1.15 + 1.32i)7-s + (0.0144 + 0.100i)8-s + (−0.319 − 0.0939i)9-s + (0.406 − 0.469i)10-s + (−1.49 − 0.961i)11-s + (0.450 + 0.289i)12-s + (0.184 − 0.212i)13-s + (2.34 + 0.687i)14-s + (0.0367 + 0.255i)15-s + (0.699 + 0.806i)16-s + (0.152 + 0.333i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.158i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 + 0.158i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.52820 - 0.202164i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.52820 - 0.202164i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.142 - 0.989i)T \) |
| 5 | \( 1 + (-0.959 + 0.281i)T \) |
| 23 | \( 1 + (4.76 + 0.569i)T \) |
| good | 2 | \( 1 + (-1.65 + 1.06i)T + (0.830 - 1.81i)T^{2} \) |
| 7 | \( 1 + (-3.04 - 3.51i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (4.96 + 3.18i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-0.663 + 0.765i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.628 - 1.37i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-2.91 + 6.37i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (0.102 + 0.225i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (1.30 + 9.10i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (1.30 + 0.382i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (10.8 - 3.18i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (0.399 - 2.77i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 - 1.05T + 47T^{2} \) |
| 53 | \( 1 + (0.978 + 1.12i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (0.884 - 1.02i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (-1.23 - 8.57i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (-5.95 + 3.82i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (-0.154 + 0.0994i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (5.74 - 12.5i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-7.38 + 8.52i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-0.184 - 0.0541i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (0.573 - 3.99i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (7.78 - 2.28i)T + (81.6 - 52.4i)T^{2} \) |
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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
−11.40556993047028526630226434380, −11.04682340685737046920365492863, −9.935813460879453362033517609811, −8.635645358444581130701005146870, −8.038305121674628430249795321877, −5.81444436384135513396931177974, −5.41145632772004380525887525442, −4.59057314582473893049349611608, −3.05970392756179531604787000617, −2.20077538556670719887688176206,
1.70380801136235924059763266934, 3.59686904771364432966993972681, 4.86592516438803101670973469976, 5.43237787407465830115374515475, 6.75678769326290751036755041091, 7.50361820256912559892514598042, 8.084195496670567377777744634388, 10.04460361674771004213660752807, 10.53908878222456222132831335711, 11.91214279716265461425545034582
