| L(s) = 1 | + (1.14 − 0.824i)2-s + (0.760 − 1.83i)3-s + (0.640 − 1.89i)4-s + (0.453 − 0.891i)5-s + (−0.640 − 2.73i)6-s + (0.369 + 1.53i)7-s + (−0.825 − 2.70i)8-s + (−0.674 − 0.674i)9-s + (−0.212 − 1.39i)10-s + (2.97 − 3.47i)11-s + (−2.99 − 2.61i)12-s + (−0.694 − 1.13i)13-s + (1.69 + 1.46i)14-s + (−1.29 − 1.51i)15-s + (−3.17 − 2.42i)16-s + (0.612 + 7.78i)17-s + ⋯ |
| L(s) = 1 | + (0.812 − 0.582i)2-s + (0.439 − 1.06i)3-s + (0.320 − 0.947i)4-s + (0.203 − 0.398i)5-s + (−0.261 − 1.11i)6-s + (0.139 + 0.581i)7-s + (−0.291 − 0.956i)8-s + (−0.224 − 0.224i)9-s + (−0.0673 − 0.442i)10-s + (0.895 − 1.04i)11-s + (−0.864 − 0.755i)12-s + (−0.192 − 0.314i)13-s + (0.452 + 0.391i)14-s + (−0.333 − 0.390i)15-s + (−0.794 − 0.606i)16-s + (0.148 + 1.88i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.665 + 0.746i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.665 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.23119 - 2.74794i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.23119 - 2.74794i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.14 + 0.824i)T \) |
| 5 | \( 1 + (-0.453 + 0.891i)T \) |
| 41 | \( 1 + (0.182 + 6.40i)T \) |
| good | 3 | \( 1 + (-0.760 + 1.83i)T + (-2.12 - 2.12i)T^{2} \) |
| 7 | \( 1 + (-0.369 - 1.53i)T + (-6.23 + 3.17i)T^{2} \) |
| 11 | \( 1 + (-2.97 + 3.47i)T + (-1.72 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.694 + 1.13i)T + (-5.90 + 11.5i)T^{2} \) |
| 17 | \( 1 + (-0.612 - 7.78i)T + (-16.7 + 2.65i)T^{2} \) |
| 19 | \( 1 + (0.189 + 0.116i)T + (8.62 + 16.9i)T^{2} \) |
| 23 | \( 1 + (4.85 + 3.52i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (0.394 - 5.00i)T + (-28.6 - 4.53i)T^{2} \) |
| 31 | \( 1 + (2.03 - 6.27i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (0.663 + 2.04i)T + (-29.9 + 21.7i)T^{2} \) |
| 43 | \( 1 + (-1.57 - 9.95i)T + (-40.8 + 13.2i)T^{2} \) |
| 47 | \( 1 + (-0.184 + 0.769i)T + (-41.8 - 21.3i)T^{2} \) |
| 53 | \( 1 + (-6.32 - 0.497i)T + (52.3 + 8.29i)T^{2} \) |
| 59 | \( 1 + (3.71 - 5.10i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (0.117 - 0.740i)T + (-58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (2.25 - 1.92i)T + (10.4 - 66.1i)T^{2} \) |
| 71 | \( 1 + (3.14 + 2.68i)T + (11.1 + 70.1i)T^{2} \) |
| 73 | \( 1 + (8.46 - 8.46i)T - 73iT^{2} \) |
| 79 | \( 1 + (2.31 + 0.958i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 - 9.11iT - 83T^{2} \) |
| 89 | \( 1 + (-7.63 + 1.83i)T + (79.2 - 40.4i)T^{2} \) |
| 97 | \( 1 + (-3.16 + 2.70i)T + (15.1 - 95.8i)T^{2} \) |
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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
−10.16299684350407412736402225030, −8.799991879016657662129847165440, −8.468837806010675495651061940414, −7.18026091650883389843589345053, −6.17138065858309943284650051730, −5.68805890809263672755842849648, −4.33267801618830724512111838039, −3.27745524368031666535880919908, −2.07158115291596674122125238550, −1.23819159716488580781935178129,
2.27992276010301900113959423721, 3.53051219527912309458857436915, 4.25084235309312229170542905597, 4.92339618758095448523791394983, 6.15013290421916339921027808983, 7.16328503520741698540405046005, 7.63739635068670401331082109221, 9.070835701363549190614561516276, 9.601654299533345686526542378987, 10.37247305612492157128125693809
