| L(s) = 1 | + (−0.862 − 0.505i)2-s + (−0.996 − 0.311i)3-s + (0.489 + 0.872i)4-s + (−0.305 + 1.77i)5-s + (0.702 + 0.772i)6-s + (0.166 + 0.498i)7-s + (0.0189 − 0.999i)8-s + (−1.57 − 1.08i)9-s + (1.16 − 1.38i)10-s + (0.0502 − 0.884i)11-s + (−0.215 − 1.02i)12-s + (−4.08 − 1.10i)13-s + (0.108 − 0.514i)14-s + (0.858 − 1.67i)15-s + (−0.521 + 0.853i)16-s + (−1.27 − 3.03i)17-s + ⋯ |
| L(s) = 1 | + (−0.610 − 0.357i)2-s + (−0.575 − 0.179i)3-s + (0.244 + 0.436i)4-s + (−0.136 + 0.795i)5-s + (0.286 + 0.315i)6-s + (0.0628 + 0.188i)7-s + (0.00669 − 0.353i)8-s + (−0.523 − 0.362i)9-s + (0.367 − 0.436i)10-s + (0.0151 − 0.266i)11-s + (−0.0622 − 0.294i)12-s + (−1.13 − 0.307i)13-s + (0.0290 − 0.137i)14-s + (0.221 − 0.433i)15-s + (−0.130 + 0.213i)16-s + (−0.308 − 0.735i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00251495 + 0.0396994i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00251495 + 0.0396994i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.862 + 0.505i)T \) |
| 167 | \( 1 + (12.0 + 4.57i)T \) |
| good | 3 | \( 1 + (0.996 + 0.311i)T + (2.46 + 1.70i)T^{2} \) |
| 5 | \( 1 + (0.305 - 1.77i)T + (-4.71 - 1.67i)T^{2} \) |
| 7 | \( 1 + (-0.166 - 0.498i)T + (-5.60 + 4.19i)T^{2} \) |
| 11 | \( 1 + (-0.0502 + 0.884i)T + (-10.9 - 1.24i)T^{2} \) |
| 13 | \( 1 + (4.08 + 1.10i)T + (11.2 + 6.57i)T^{2} \) |
| 17 | \( 1 + (1.27 + 3.03i)T + (-11.9 + 12.1i)T^{2} \) |
| 19 | \( 1 + (1.68 + 0.822i)T + (11.6 + 14.9i)T^{2} \) |
| 23 | \( 1 + (7.95 - 3.51i)T + (15.4 - 17.0i)T^{2} \) |
| 29 | \( 1 + (8.07 + 3.57i)T + (19.5 + 21.4i)T^{2} \) |
| 31 | \( 1 + (-1.58 + 0.630i)T + (22.5 - 21.2i)T^{2} \) |
| 37 | \( 1 + (5.84 - 4.04i)T + (13.0 - 34.6i)T^{2} \) |
| 41 | \( 1 + (2.49 - 2.36i)T + (2.32 - 40.9i)T^{2} \) |
| 43 | \( 1 + (-2.92 - 2.37i)T + (8.88 + 42.0i)T^{2} \) |
| 47 | \( 1 + (4.03 + 2.16i)T + (26.0 + 39.1i)T^{2} \) |
| 53 | \( 1 + (2.66 + 9.13i)T + (-44.6 + 28.5i)T^{2} \) |
| 59 | \( 1 + (2.15 - 5.13i)T + (-41.3 - 42.1i)T^{2} \) |
| 61 | \( 1 + (-2.77 + 11.0i)T + (-53.7 - 28.8i)T^{2} \) |
| 67 | \( 1 + (-1.70 - 9.89i)T + (-63.1 + 22.3i)T^{2} \) |
| 71 | \( 1 + (-4.10 + 0.786i)T + (65.9 - 26.2i)T^{2} \) |
| 73 | \( 1 + (1.93 + 3.16i)T + (-33.2 + 64.9i)T^{2} \) |
| 79 | \( 1 + (-15.4 - 1.17i)T + (78.0 + 11.9i)T^{2} \) |
| 83 | \( 1 + (8.07 - 4.73i)T + (40.5 - 72.3i)T^{2} \) |
| 89 | \( 1 + (-1.50 - 2.94i)T + (-52.0 + 72.2i)T^{2} \) |
| 97 | \( 1 + (-4.64 - 1.84i)T + (70.5 + 66.6i)T^{2} \) |
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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
−11.26472602460600254317472543951, −10.18905223258397344085568972408, −9.414261878584136800945875439842, −8.236516005311753412634972525860, −7.25739398871403276452922282008, −6.37375871410145265421826092340, −5.22611383162453156588560036110, −3.52360285298388318265375745916, −2.31743121922312073954223488370, −0.03371578345849030252214087909,
2.06287499936268107195555854037, 4.26687094781197803153387635316, 5.22545943895526725421416042804, 6.20499247872842008968300079249, 7.41097086341672710590512977546, 8.342132061352627075835891439226, 9.155143207898469318648447677775, 10.25425433208388023271845384359, 10.90160201960093013068423721525, 12.07654947331746046688662961430
