| L(s) = 1 | + (−0.0523 + 0.998i)2-s + (−0.879 − 2.29i)3-s + (−0.994 − 0.104i)4-s + (0.496 − 2.18i)5-s + (2.33 − 0.758i)6-s + (−2.23 − 1.40i)7-s + (0.156 − 0.987i)8-s + (−2.24 + 2.02i)9-s + (2.15 + 0.609i)10-s + (2.96 − 1.48i)11-s + (0.635 + 2.37i)12-s + (1.00 − 0.510i)13-s + (1.52 − 2.16i)14-s + (−5.43 + 0.780i)15-s + (0.978 + 0.207i)16-s + (−0.133 − 2.54i)17-s + ⋯ |
| L(s) = 1 | + (−0.0370 + 0.706i)2-s + (−0.507 − 1.32i)3-s + (−0.497 − 0.0522i)4-s + (0.221 − 0.975i)5-s + (0.952 − 0.309i)6-s + (−0.846 − 0.532i)7-s + (0.0553 − 0.349i)8-s + (−0.748 + 0.674i)9-s + (0.680 + 0.192i)10-s + (0.893 − 0.449i)11-s + (0.183 + 0.684i)12-s + (0.278 − 0.141i)13-s + (0.407 − 0.578i)14-s + (−1.40 + 0.201i)15-s + (0.244 + 0.0519i)16-s + (−0.0323 − 0.617i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.915 + 0.403i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.915 + 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.166779 - 0.791833i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.166779 - 0.791833i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0523 - 0.998i)T \) |
| 5 | \( 1 + (-0.496 + 2.18i)T \) |
| 7 | \( 1 + (2.23 + 1.40i)T \) |
| 11 | \( 1 + (-2.96 + 1.48i)T \) |
| good | 3 | \( 1 + (0.879 + 2.29i)T + (-2.22 + 2.00i)T^{2} \) |
| 13 | \( 1 + (-1.00 + 0.510i)T + (7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (0.133 + 2.54i)T + (-16.9 + 1.77i)T^{2} \) |
| 19 | \( 1 + (0.0883 + 0.840i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (-0.490 - 1.82i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (3.35 - 4.61i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (0.824 + 3.87i)T + (-28.3 + 12.6i)T^{2} \) |
| 37 | \( 1 + (7.12 + 2.73i)T + (27.4 + 24.7i)T^{2} \) |
| 41 | \( 1 + (-6.45 - 8.88i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (5.01 - 5.01i)T - 43iT^{2} \) |
| 47 | \( 1 + (2.51 + 2.03i)T + (9.77 + 45.9i)T^{2} \) |
| 53 | \( 1 + (3.73 - 5.75i)T + (-21.5 - 48.4i)T^{2} \) |
| 59 | \( 1 + (-0.430 + 4.09i)T + (-57.7 - 12.2i)T^{2} \) |
| 61 | \( 1 + (1.94 - 9.14i)T + (-55.7 - 24.8i)T^{2} \) |
| 67 | \( 1 + (-2.70 + 10.1i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (3.11 + 9.58i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-8.58 + 6.95i)T + (15.1 - 71.4i)T^{2} \) |
| 79 | \( 1 + (-4.23 + 3.80i)T + (8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (-6.46 + 12.6i)T + (-48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + (-0.804 + 1.39i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.89 - 11.5i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−9.557670255143604057212869629232, −9.081269375486718098715862506077, −7.974542001427695947166104763555, −7.24966956009022544743129324761, −6.39687645163340253895902501984, −5.89595679151791678678424265241, −4.76881793905528913700119092739, −3.48514270484600698979275343665, −1.51522159637065025979800223796, −0.45862082951768425608353472940,
2.13043610037922264212731722919, 3.49665661565733524809645951246, 3.95150081445558920908042437907, 5.25337085528419421047949076312, 6.13870514697391075616585926092, 6.99011712960455493310824825262, 8.571464272906453631766714165685, 9.435147637865602716295310004300, 9.945074184238437209826527568640, 10.58756660006493162650720141736
