L(s) = 1 | + (−0.866 + 1.5i)2-s + (−1.36 + 2.36i)3-s + (−0.5 − 0.866i)4-s + 1.73·5-s + (−2.36 − 4.09i)6-s + (−0.5 − 0.866i)7-s − 1.73·8-s + (−2.23 − 3.86i)9-s + (−1.49 + 2.59i)10-s + (−0.633 + 1.09i)11-s + 2.73·12-s + (3.59 − 0.232i)13-s + 1.73·14-s + (−2.36 + 4.09i)15-s + (2.49 − 4.33i)16-s + (3.86 + 6.69i)17-s + ⋯ |
L(s) = 1 | + (−0.612 + 1.06i)2-s + (−0.788 + 1.36i)3-s + (−0.250 − 0.433i)4-s + 0.774·5-s + (−0.965 − 1.67i)6-s + (−0.188 − 0.327i)7-s − 0.612·8-s + (−0.744 − 1.28i)9-s + (−0.474 + 0.821i)10-s + (−0.191 + 0.331i)11-s + 0.788·12-s + (0.997 − 0.0643i)13-s + 0.462·14-s + (−0.610 + 1.05i)15-s + (0.624 − 1.08i)16-s + (0.937 + 1.62i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.202i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.979 - 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0634710 + 0.620695i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0634710 + 0.620695i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 + (-3.59 + 0.232i)T \) |
good | 2 | \( 1 + (0.866 - 1.5i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (1.36 - 2.36i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 - 1.73T + 5T^{2} \) |
| 11 | \( 1 + (0.633 - 1.09i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-3.86 - 6.69i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.36 - 4.09i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 4.19T + 31T^{2} \) |
| 37 | \( 1 + (-3.5 + 6.06i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.59 + 4.5i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.0980 - 0.169i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 12.9T + 47T^{2} \) |
| 53 | \( 1 + 9.92T + 53T^{2} \) |
| 59 | \( 1 + (3.63 + 6.29i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.40 - 4.16i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.09 + 5.36i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 3.19T + 73T^{2} \) |
| 79 | \( 1 - 16.1T + 79T^{2} \) |
| 83 | \( 1 + 2.19T + 83T^{2} \) |
| 89 | \( 1 + (6.46 - 11.1i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (3.19 + 5.53i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−15.15666101548764240559220355731, −13.88196892102361452841898463818, −12.37968902497733386537962002930, −10.93689010970627290765972950814, −10.04946838483038308866525894690, −9.253876937203867667498890562736, −7.924442681759987638474619549944, −6.21385964025188172171290512662, −5.64231538960605410821936467083, −3.88086092302963128809729415964,
1.12534378376491676230474080064, 2.64588382454827270263349499521, 5.69536135995479125566446277246, 6.46336141615095059662223825229, 8.090634438141046613290676055716, 9.427670507004874679649652828066, 10.52807817338731503102535610609, 11.60417710998818345196829919010, 12.23005857609586705230924641082, 13.24586092230948158654969932437