L(s) = 1 | + (0.866 − 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 − 0.866i)4-s + (−0.499 + 0.866i)6-s + (−3.08 − 1.78i)7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (2.06 + 3.57i)11-s + 0.999i·12-s + (1.35 + 3.34i)13-s − 3.56·14-s + (−0.5 − 0.866i)16-s + (−4.43 − 2.56i)17-s − 0.999i·18-s + (−1.78 + 3.08i)19-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (−0.499 + 0.288i)3-s + (0.249 − 0.433i)4-s + (−0.204 + 0.353i)6-s + (−1.16 − 0.673i)7-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (0.621 + 1.07i)11-s + 0.288i·12-s + (0.375 + 0.926i)13-s − 0.951·14-s + (−0.125 − 0.216i)16-s + (−1.07 − 0.621i)17-s − 0.235i·18-s + (−0.408 + 0.707i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.187i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 - 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.780160217\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.780160217\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-1.35 - 3.34i)T \) |
good | 7 | \( 1 + (3.08 + 1.78i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.06 - 3.57i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (4.43 + 2.56i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.78 - 3.08i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.65 + 3.84i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.28 - 5.68i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 5.68T + 31T^{2} \) |
| 37 | \( 1 + (-3.57 + 2.06i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.12 - 3.67i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.95 - 2.28i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 7iT - 47T^{2} \) |
| 53 | \( 1 + 4.43iT - 53T^{2} \) |
| 59 | \( 1 + (-5.28 + 9.14i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3 - 5.19i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-12.3 + 7.12i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.43 + 4.22i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 15.3iT - 73T^{2} \) |
| 79 | \( 1 + 7.43T + 79T^{2} \) |
| 83 | \( 1 + 1.12iT - 83T^{2} \) |
| 89 | \( 1 + (-0.903 - 1.56i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.972 + 0.561i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.522789402914728436577702557522, −8.681799137533102319359082144877, −7.17184241461247576083681623445, −6.65221065272230693576379007909, −6.24494153011246076732337958845, −4.80309213147211005861243114224, −4.39170521973504649722558090999, −3.52711322970781169105374422348, −2.41755616200565892754369870735, −1.01106139903692672474078142961,
0.70322667241907149127332760433, 2.51184252108664876834001200849, 3.28412584279150506615181350224, 4.27472756660577133447272041928, 5.37607643544604316464862309746, 6.14140497595238526512965329964, 6.42863107033293858876899320496, 7.33503906850662907052735131180, 8.508297606792610297960410059675, 8.887812560935922427660816893786