| L(s) = 1 | + (−0.256 − 0.222i)2-s + (1.36 + 1.06i)3-s + (−0.268 − 1.86i)4-s + (−0.197 − 0.432i)5-s + (−0.114 − 0.577i)6-s + (0.641 + 2.18i)7-s + (−0.713 + 1.11i)8-s + (0.737 + 2.90i)9-s + (−0.0455 + 0.155i)10-s + (−3.61 − 4.16i)11-s + (1.61 − 2.83i)12-s + (−0.876 − 0.257i)13-s + (0.321 − 0.703i)14-s + (0.189 − 0.801i)15-s + (−3.18 + 0.935i)16-s + (−0.637 + 4.43i)17-s + ⋯ |
| L(s) = 1 | + (−0.181 − 0.157i)2-s + (0.789 + 0.614i)3-s + (−0.134 − 0.932i)4-s + (−0.0883 − 0.193i)5-s + (−0.0467 − 0.235i)6-s + (0.242 + 0.825i)7-s + (−0.252 + 0.392i)8-s + (0.245 + 0.969i)9-s + (−0.0143 + 0.0490i)10-s + (−1.08 − 1.25i)11-s + (0.466 − 0.818i)12-s + (−0.243 − 0.0713i)13-s + (0.0859 − 0.188i)14-s + (0.0490 − 0.206i)15-s + (−0.796 + 0.233i)16-s + (−0.154 + 1.07i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.111i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.966406 - 0.0541909i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.966406 - 0.0541909i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.36 - 1.06i)T \) |
| 23 | \( 1 + (-2.58 + 4.04i)T \) |
| good | 2 | \( 1 + (0.256 + 0.222i)T + (0.284 + 1.97i)T^{2} \) |
| 5 | \( 1 + (0.197 + 0.432i)T + (-3.27 + 3.77i)T^{2} \) |
| 7 | \( 1 + (-0.641 - 2.18i)T + (-5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (3.61 + 4.16i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.876 + 0.257i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (0.637 - 4.43i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (3.96 - 0.570i)T + (18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-2.58 - 0.372i)T + (27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-2.56 - 1.65i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (-1.02 - 0.467i)T + (24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (-5.22 + 2.38i)T + (26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (1.87 + 2.91i)T + (-17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 - 10.1iT - 47T^{2} \) |
| 53 | \( 1 + (-6.20 + 1.82i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (-1.57 + 5.37i)T + (-49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (2.10 - 3.27i)T + (-25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (7.57 + 6.56i)T + (9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (9.28 + 8.04i)T + (10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (1.61 + 11.2i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (4.00 - 13.6i)T + (-66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (3.08 - 6.74i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (12.2 - 7.87i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-2.94 + 1.34i)T + (63.5 - 73.3i)T^{2} \) |
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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
−14.84758482923822023543760277032, −13.88833392152222387422287301551, −12.68727081856388203801733154165, −10.91488598212432736351029204370, −10.33311933118098207012025756316, −8.828323735774503762198649477475, −8.334714831539520322237460802352, −5.97528392062912277463272743575, −4.72210913658375747274540856224, −2.56864543724664299372288758053,
2.73817656020984776452418788453, 4.43511812176085237160784481942, 7.15540753751905382284380577766, 7.46742768832970676950610162005, 8.787464490625036244865074959848, 10.04741106606674171602057852968, 11.67590896023618377367846074811, 12.90293396425365113512585793517, 13.43328279623955285782907996490, 14.75683602266193290638750913982
