| L(s) = 1 | + (0.0204 + 0.0314i)2-s + (−0.0391 − 0.102i)3-s + (0.812 − 1.82i)4-s + (−2.17 + 0.523i)5-s + (0.00240 − 0.00331i)6-s + (1.46 − 2.20i)7-s + (0.147 − 0.0234i)8-s + (2.22 − 1.99i)9-s + (−0.0608 − 0.0576i)10-s + (0.920 − 1.02i)11-s + (−0.218 − 0.0114i)12-s + (−1.43 + 0.730i)13-s + (0.0991 + 0.00105i)14-s + (0.138 + 0.201i)15-s + (−2.67 − 2.96i)16-s + (2.74 + 3.39i)17-s + ⋯ |
| L(s) = 1 | + (0.0144 + 0.0222i)2-s + (−0.0226 − 0.0589i)3-s + (0.406 − 0.912i)4-s + (−0.972 + 0.234i)5-s + (0.000982 − 0.00135i)6-s + (0.553 − 0.832i)7-s + (0.0523 − 0.00828i)8-s + (0.740 − 0.666i)9-s + (−0.0192 − 0.0182i)10-s + (0.277 − 0.308i)11-s + (−0.0629 − 0.00329i)12-s + (−0.397 + 0.202i)13-s + (0.0264 + 0.000282i)14-s + (0.0357 + 0.0519i)15-s + (−0.667 − 0.741i)16-s + (0.666 + 0.823i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.526 + 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.526 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.01731 - 0.566204i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01731 - 0.566204i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (2.17 - 0.523i)T \) |
| 7 | \( 1 + (-1.46 + 2.20i)T \) |
| good | 2 | \( 1 + (-0.0204 - 0.0314i)T + (-0.813 + 1.82i)T^{2} \) |
| 3 | \( 1 + (0.0391 + 0.102i)T + (-2.22 + 2.00i)T^{2} \) |
| 11 | \( 1 + (-0.920 + 1.02i)T + (-1.14 - 10.9i)T^{2} \) |
| 13 | \( 1 + (1.43 - 0.730i)T + (7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (-2.74 - 3.39i)T + (-3.53 + 16.6i)T^{2} \) |
| 19 | \( 1 + (0.0625 - 0.0278i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (4.73 - 3.07i)T + (9.35 - 21.0i)T^{2} \) |
| 29 | \( 1 + (-3.88 - 5.35i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-3.48 - 0.366i)T + (30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (0.405 - 7.73i)T + (-36.7 - 3.86i)T^{2} \) |
| 41 | \( 1 + (9.12 + 2.96i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (3.38 - 3.38i)T - 43iT^{2} \) |
| 47 | \( 1 + (-5.54 - 4.48i)T + (9.77 + 45.9i)T^{2} \) |
| 53 | \( 1 + (2.76 - 1.06i)T + (39.3 - 35.4i)T^{2} \) |
| 59 | \( 1 + (-10.0 + 2.13i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (1.57 - 7.39i)T + (-55.7 - 24.8i)T^{2} \) |
| 67 | \( 1 + (-6.89 + 5.58i)T + (13.9 - 65.5i)T^{2} \) |
| 71 | \( 1 + (-5.52 + 4.01i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (0.428 - 0.0224i)T + (72.6 - 7.63i)T^{2} \) |
| 79 | \( 1 + (-9.05 + 0.951i)T + (77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (0.663 + 4.19i)T + (-78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (7.87 + 1.67i)T + (81.3 + 36.1i)T^{2} \) |
| 97 | \( 1 + (-2.59 + 16.3i)T + (-92.2 - 29.9i)T^{2} \) |
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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
−12.26905500927574929145342738878, −11.57787021219516702205550454897, −10.53253183651901346310779743289, −9.875283992036883153728774800684, −8.302176643911900618255830678562, −7.22612063051556001623805203997, −6.41208415058377664081044297745, −4.80278210764708420667724507624, −3.62698479382907530612219465984, −1.28070929817424346232277145909,
2.38730972066450208382948190941, 3.99326296655437301698336813507, 5.06564773126139187991968858378, 6.89540514862470197489920721965, 7.86514396550820082383723488657, 8.448770331337725215302543638979, 9.907759067522806786193845496476, 11.21059509807892452401731044730, 12.08265856038936589972493710615, 12.41413474923283786264589350260
