| L(s) = 1 | + (−0.557 − 0.405i)2-s + (0.824 + 0.367i)3-s + (−0.471 − 1.44i)4-s + (−0.311 − 0.538i)6-s + (−0.510 + 0.567i)7-s + (−0.750 + 2.31i)8-s + (−1.46 − 1.62i)9-s + (4.02 + 0.855i)11-s + (0.143 − 1.36i)12-s + (−0.304 − 2.89i)13-s + (0.514 − 0.109i)14-s + (−1.11 + 0.807i)16-s + (1.27 − 0.272i)17-s + (0.157 + 1.49i)18-s + (0.397 − 3.77i)19-s + ⋯ |
| L(s) = 1 | + (−0.394 − 0.286i)2-s + (0.475 + 0.211i)3-s + (−0.235 − 0.724i)4-s + (−0.127 − 0.219i)6-s + (−0.193 + 0.214i)7-s + (−0.265 + 0.817i)8-s + (−0.487 − 0.541i)9-s + (1.21 + 0.257i)11-s + (0.0415 − 0.394i)12-s + (−0.0843 − 0.802i)13-s + (0.137 − 0.0292i)14-s + (−0.277 + 0.201i)16-s + (0.310 − 0.0659i)17-s + (0.0371 + 0.353i)18-s + (0.0911 − 0.866i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.372 + 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.372 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.621673 - 0.919285i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.621673 - 0.919285i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 31 | \( 1 + (1.63 + 5.32i)T \) |
| good | 2 | \( 1 + (0.557 + 0.405i)T + (0.618 + 1.90i)T^{2} \) |
| 3 | \( 1 + (-0.824 - 0.367i)T + (2.00 + 2.22i)T^{2} \) |
| 7 | \( 1 + (0.510 - 0.567i)T + (-0.731 - 6.96i)T^{2} \) |
| 11 | \( 1 + (-4.02 - 0.855i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (0.304 + 2.89i)T + (-12.7 + 2.70i)T^{2} \) |
| 17 | \( 1 + (-1.27 + 0.272i)T + (15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (-0.397 + 3.77i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (-1.01 + 3.12i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (3.96 + 2.87i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 + (5.20 + 9.01i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.690 + 0.307i)T + (27.4 - 30.4i)T^{2} \) |
| 43 | \( 1 + (0.748 - 7.11i)T + (-42.0 - 8.94i)T^{2} \) |
| 47 | \( 1 + (0.708 - 0.514i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (2.39 + 2.65i)T + (-5.54 + 52.7i)T^{2} \) |
| 59 | \( 1 + (-0.847 - 0.377i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 - 2.31T + 61T^{2} \) |
| 67 | \( 1 + (1.04 - 1.80i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (5.17 + 5.74i)T + (-7.42 + 70.6i)T^{2} \) |
| 73 | \( 1 + (5.53 + 1.17i)T + (66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (-13.7 + 2.93i)T + (72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (-12.8 + 5.73i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (1.37 + 4.21i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-1.63 - 5.01i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−9.785801712060530160407710963595, −9.250730000564705822808968065542, −8.749607031448025542659359398863, −7.63487108638886353638974071991, −6.38752187836853076629826607636, −5.70059528222972963815415603813, −4.52164109311308117149797249269, −3.36446364406141735124973015595, −2.19361984579352800563046639078, −0.62132708318879952688338748056,
1.64353470928658298952405507624, 3.25446973475181796036392841617, 3.87782795148304560791986030026, 5.24155324423520421343312722089, 6.55628075495397742387607382297, 7.21352003592393866651645490359, 8.103819262663060641686543057202, 8.828163369604463695669948423199, 9.369594296727680042785326771863, 10.40737385671226060838062128909
