| L(s) = 1 | + (0.315 − 1.37i)2-s + (−1.80 − 0.868i)4-s + (0.437 + 0.159i)5-s + (−3.46 − 0.610i)7-s + (−1.76 + 2.21i)8-s + (0.357 − 0.552i)10-s + (−0.485 − 1.33i)11-s + (−1.62 − 1.93i)13-s + (−1.93 + 4.58i)14-s + (2.49 + 3.12i)16-s + (0.667 − 0.385i)17-s + (−3.66 + 6.34i)19-s + (−0.649 − 0.666i)20-s + (−1.99 + 0.249i)22-s + (1.25 + 7.12i)23-s + ⋯ |
| L(s) = 1 | + (0.222 − 0.974i)2-s + (−0.900 − 0.434i)4-s + (0.195 + 0.0711i)5-s + (−1.30 − 0.230i)7-s + (−0.624 + 0.781i)8-s + (0.112 − 0.174i)10-s + (−0.146 − 0.402i)11-s + (−0.449 − 0.535i)13-s + (−0.516 + 1.22i)14-s + (0.622 + 0.782i)16-s + (0.161 − 0.0934i)17-s + (−0.839 + 1.45i)19-s + (−0.145 − 0.149i)20-s + (−0.425 + 0.0531i)22-s + (0.261 + 1.48i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.409 - 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.409 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0702067 + 0.108458i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0702067 + 0.108458i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.315 + 1.37i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-0.437 - 0.159i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (3.46 + 0.610i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (0.485 + 1.33i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (1.62 + 1.93i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.667 + 0.385i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.66 - 6.34i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.25 - 7.12i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (3.15 + 2.64i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (7.80 - 1.37i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (2.79 - 1.61i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.74 + 2.07i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (1.99 - 0.726i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.82 + 10.3i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 8.73T + 53T^{2} \) |
| 59 | \( 1 + (0.818 - 2.24i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-3.02 - 0.533i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (4.04 - 3.39i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (5.70 + 9.88i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.96 + 5.12i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.94 + 10.6i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (7.41 - 8.83i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (8.59 + 4.96i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.95 - 0.710i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.08427722656839940002068968207, −9.459924117166106111467899156924, −8.458035113131889275733039358321, −7.33858461242303974986095585675, −6.02805912787721765201558756899, −5.43696602722016416649831588241, −3.87165259710787591521720316846, −3.29342527228168236320308387548, −1.94363329894808626000404113062, −0.06096080930938723741635039079,
2.57105086964427760666729935673, 3.84222252185133951029314711492, 4.87727860028512317364871516230, 5.89314675749352045575969583089, 6.77419902658980559630805610661, 7.29037319120715506384206552356, 8.653214685546220013015259562080, 9.259950784694153764850969848045, 9.933952542132831639741713049479, 11.08973587497913863637563837045
