L(s) = 1 | + (−2.49 − 0.668i)2-s + (0.116 − 1.72i)3-s + (4.03 + 2.33i)4-s + (−0.624 + 0.624i)5-s + (−1.44 + 4.23i)6-s + (0.435 + 1.62i)7-s + (−4.86 − 4.86i)8-s + (−2.97 − 0.401i)9-s + (1.97 − 1.14i)10-s + (0.0927 − 0.346i)11-s + (4.49 − 6.70i)12-s − 4.34i·14-s + (1.00 + 1.15i)15-s + (4.20 + 7.29i)16-s + (−1.31 + 2.27i)17-s + (7.14 + 2.98i)18-s + ⋯ |
L(s) = 1 | + (−1.76 − 0.472i)2-s + (0.0669 − 0.997i)3-s + (2.01 + 1.16i)4-s + (−0.279 + 0.279i)5-s + (−0.589 + 1.72i)6-s + (0.164 + 0.613i)7-s + (−1.71 − 1.71i)8-s + (−0.991 − 0.133i)9-s + (0.624 − 0.360i)10-s + (0.0279 − 0.104i)11-s + (1.29 − 1.93i)12-s − 1.15i·14-s + (0.260 + 0.297i)15-s + (1.05 + 1.82i)16-s + (−0.318 + 0.551i)17-s + (1.68 + 0.703i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 + 0.565i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.824 + 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.564563 - 0.175012i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.564563 - 0.175012i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.116 + 1.72i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (2.49 + 0.668i)T + (1.73 + i)T^{2} \) |
| 5 | \( 1 + (0.624 - 0.624i)T - 5iT^{2} \) |
| 7 | \( 1 + (-0.435 - 1.62i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-0.0927 + 0.346i)T + (-9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (1.31 - 2.27i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.90 + 1.58i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-2.70 - 4.69i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-7.24 + 4.18i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1.27 + 1.27i)T + 31iT^{2} \) |
| 37 | \( 1 + (3.33 + 0.894i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-6.68 - 1.79i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-0.825 - 0.476i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.33 - 5.33i)T + 47iT^{2} \) |
| 53 | \( 1 + 9.69iT - 53T^{2} \) |
| 59 | \( 1 + (-9.95 + 2.66i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (2.43 - 4.22i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.25 + 12.1i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-0.832 - 3.10i)T + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (0.246 - 0.246i)T - 73iT^{2} \) |
| 79 | \( 1 + 3.68T + 79T^{2} \) |
| 83 | \( 1 + (8.47 - 8.47i)T - 83iT^{2} \) |
| 89 | \( 1 + (1.77 - 6.61i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (-5.11 + 1.36i)T + (84.0 - 48.5i)T^{2} \) |
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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
−10.95398311958809176930347381153, −9.719739449419776342547907070334, −8.997546607967832339590748736623, −8.206082626657161078501726958690, −7.48151299971136171506542627379, −6.77948873041997832774520122550, −5.59629345998732211398418226552, −3.26789947474968707083353485518, −2.28699963950553822701778491412, −1.03478059231956552484473105839,
0.816674428292540503400727423591, 2.75964444919661153376619453306, 4.36915374621764754447063819172, 5.53593026862762296989031043641, 6.80245205564414179608609747234, 7.61797291370386769848551699673, 8.596286060086469145024051624786, 9.069350094693824918453837623039, 10.11811799185882140536024834580, 10.49410892463238390517425187565