L(s) = 1 | + (−1.02 − 0.593i)2-s + (0.298 + 0.172i)3-s + (−0.295 − 0.511i)4-s + (1.44 − 1.71i)5-s + (−0.204 − 0.354i)6-s + (1.75 − 1.01i)7-s + 3.07i·8-s + (−1.44 − 2.49i)9-s + (−2.49 + 0.903i)10-s + (−1.94 + 3.36i)11-s − 0.203i·12-s + (2.96 + 2.05i)13-s − 2.40·14-s + (0.725 − 0.262i)15-s + (1.23 − 2.14i)16-s + (−4.71 + 2.72i)17-s + ⋯ |
L(s) = 1 | + (−0.727 − 0.419i)2-s + (0.172 + 0.0996i)3-s + (−0.147 − 0.255i)4-s + (0.644 − 0.764i)5-s + (−0.0836 − 0.144i)6-s + (0.664 − 0.383i)7-s + 1.08i·8-s + (−0.480 − 0.831i)9-s + (−0.789 + 0.285i)10-s + (−0.585 + 1.01i)11-s − 0.0588i·12-s + (0.821 + 0.570i)13-s − 0.644·14-s + (0.187 − 0.0678i)15-s + (0.308 − 0.535i)16-s + (−1.14 + 0.660i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.496 + 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.496 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.604396 - 0.350547i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.604396 - 0.350547i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-1.44 + 1.71i)T \) |
| 13 | \( 1 + (-2.96 - 2.05i)T \) |
good | 2 | \( 1 + (1.02 + 0.593i)T + (1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.298 - 0.172i)T + (1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + (-1.75 + 1.01i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.94 - 3.36i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (4.71 - 2.72i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.94 - 5.09i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.298 + 0.172i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 1.18T + 31T^{2} \) |
| 37 | \( 1 + (4.71 + 2.72i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.0902 - 0.156i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.15 + 0.669i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 12.2iT - 47T^{2} \) |
| 53 | \( 1 + 2.42iT - 53T^{2} \) |
| 59 | \( 1 + (3.53 + 6.11i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.38 + 5.85i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.81 - 2.20i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.940 - 1.62i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 8.86iT - 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 + 7.83iT - 83T^{2} \) |
| 89 | \( 1 + (-6.12 + 10.6i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (5.02 - 2.90i)T + (48.5 - 84.0i)T^{2} \) |
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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
−14.55636927602457251957202381864, −13.75333028631754246437917039593, −12.38693891466126190098742838451, −11.10612773286211383650830925362, −9.978732064585027562161482272100, −9.073303178768509837174275897747, −8.119972329407518436694905877159, −6.02274002914430513583942533406, −4.53893223954281904920732777125, −1.71459040122033398318564720035,
2.88781723689670296305944389893, 5.36259536706706809649376147535, 6.93057783513284253217187096663, 8.210623124054491810145011697738, 8.971942702981879194484252822984, 10.53529792906180371488278811811, 11.39880010289658299067146208597, 13.41262942952275426455489460140, 13.71018304155990256879656864479, 15.32190950894775348440988316763