| L(s) = 1 | + (−2.25 − 0.339i)2-s + (0.489 − 0.334i)3-s + (3.05 + 0.943i)4-s + (−0.260 − 3.47i)5-s + (−1.21 + 0.586i)6-s + (1.30 − 2.30i)7-s + (−2.46 − 1.18i)8-s + (−0.967 + 2.46i)9-s + (−0.594 + 7.92i)10-s + (0.728 + 1.85i)11-s + (1.81 − 0.559i)12-s + (−0.725 + 0.909i)13-s + (−3.71 + 4.75i)14-s + (−1.28 − 1.61i)15-s + (−0.128 − 0.0873i)16-s + (0.130 + 0.120i)17-s + ⋯ |
| L(s) = 1 | + (−1.59 − 0.240i)2-s + (0.282 − 0.192i)3-s + (1.52 + 0.471i)4-s + (−0.116 − 1.55i)5-s + (−0.497 + 0.239i)6-s + (0.491 − 0.870i)7-s + (−0.872 − 0.420i)8-s + (−0.322 + 0.821i)9-s + (−0.187 + 2.50i)10-s + (0.219 + 0.559i)11-s + (0.523 − 0.161i)12-s + (−0.201 + 0.252i)13-s + (−0.993 + 1.27i)14-s + (−0.332 − 0.417i)15-s + (−0.0320 − 0.0218i)16-s + (0.0315 + 0.0293i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.403 + 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.403 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.386180 - 0.251847i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.386180 - 0.251847i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-1.30 + 2.30i)T \) |
| good | 2 | \( 1 + (2.25 + 0.339i)T + (1.91 + 0.589i)T^{2} \) |
| 3 | \( 1 + (-0.489 + 0.334i)T + (1.09 - 2.79i)T^{2} \) |
| 5 | \( 1 + (0.260 + 3.47i)T + (-4.94 + 0.745i)T^{2} \) |
| 11 | \( 1 + (-0.728 - 1.85i)T + (-8.06 + 7.48i)T^{2} \) |
| 13 | \( 1 + (0.725 - 0.909i)T + (-2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (-0.130 - 0.120i)T + (1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-2.36 - 4.09i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.75 + 2.55i)T + (1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (2.03 - 8.90i)T + (-26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (-4.06 + 7.04i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.87 - 1.50i)T + (30.5 - 20.8i)T^{2} \) |
| 41 | \( 1 + (-4.57 - 2.20i)T + (25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (-1.63 + 0.787i)T + (26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (1.31 + 0.198i)T + (44.9 + 13.8i)T^{2} \) |
| 53 | \( 1 + (7.14 + 2.20i)T + (43.7 + 29.8i)T^{2} \) |
| 59 | \( 1 + (-0.0395 + 0.528i)T + (-58.3 - 8.79i)T^{2} \) |
| 61 | \( 1 + (3.38 - 1.04i)T + (50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (-0.223 + 0.387i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.323 + 1.41i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (13.2 - 1.99i)T + (69.7 - 21.5i)T^{2} \) |
| 79 | \( 1 + (-1.42 - 2.47i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.35 + 4.21i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (-2.62 + 6.67i)T + (-65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 - 1.58T + 97T^{2} \) |
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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
−16.17136822321808702752289033421, −14.28002337298141407500929296874, −12.95949125078269040493900658261, −11.67798561198835492460472145252, −10.43059309803771948460138931909, −9.226729130274934155764819385754, −8.254261057840123649968324760607, −7.43072985358662533360185229729, −4.79480385806002732508314190587, −1.49864112530701304571678929839,
2.87801472955061745114941037994, 6.22136791993410424702321765206, 7.42508488801788592083525789338, 8.700589038477517958082364801295, 9.678635673907050510856031371197, 10.93913838273404895610708493546, 11.71603868303269336517425355939, 14.06642733016359377870208393459, 15.16852084906150359844329196026, 15.69582121778202061002007323540
