L(s) = 1 | + (0.250 + 1.39i)2-s + (−1.01 + 1.40i)3-s + (−1.87 + 0.696i)4-s + (2.23 + 0.116i)5-s + (−2.20 − 1.06i)6-s + (−2.29 + 2.29i)7-s + (−1.43 − 2.43i)8-s + (−0.925 − 2.85i)9-s + (0.396 + 3.13i)10-s + 2.28·11-s + (0.934 − 3.33i)12-s + (−1.05 + 1.05i)13-s + (−3.76 − 2.61i)14-s + (−2.43 + 3.00i)15-s + (3.03 − 2.61i)16-s + (3.04 + 3.04i)17-s + ⋯ |
L(s) = 1 | + (0.176 + 0.984i)2-s + (−0.588 + 0.808i)3-s + (−0.937 + 0.348i)4-s + (0.998 + 0.0519i)5-s + (−0.900 − 0.435i)6-s + (−0.865 + 0.865i)7-s + (−0.508 − 0.861i)8-s + (−0.308 − 0.951i)9-s + (0.125 + 0.992i)10-s + 0.688·11-s + (0.269 − 0.962i)12-s + (−0.292 + 0.292i)13-s + (−1.00 − 0.698i)14-s + (−0.629 + 0.777i)15-s + (0.757 − 0.652i)16-s + (0.738 + 0.738i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.764 - 0.644i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.324170 + 0.886687i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.324170 + 0.886687i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.250 - 1.39i)T \) |
| 3 | \( 1 + (1.01 - 1.40i)T \) |
| 5 | \( 1 + (-2.23 - 0.116i)T \) |
good | 7 | \( 1 + (2.29 - 2.29i)T - 7iT^{2} \) |
| 11 | \( 1 - 2.28T + 11T^{2} \) |
| 13 | \( 1 + (1.05 - 1.05i)T - 13iT^{2} \) |
| 17 | \( 1 + (-3.04 - 3.04i)T + 17iT^{2} \) |
| 19 | \( 1 - 3.36T + 19T^{2} \) |
| 23 | \( 1 + (-3.68 + 3.68i)T - 23iT^{2} \) |
| 29 | \( 1 - 2.71iT - 29T^{2} \) |
| 31 | \( 1 + 6.49T + 31T^{2} \) |
| 37 | \( 1 + (2.31 + 2.31i)T + 37iT^{2} \) |
| 41 | \( 1 + 10.8iT - 41T^{2} \) |
| 43 | \( 1 + (-1.16 + 1.16i)T - 43iT^{2} \) |
| 47 | \( 1 + (1.83 + 1.83i)T + 47iT^{2} \) |
| 53 | \( 1 + (5.82 + 5.82i)T + 53iT^{2} \) |
| 59 | \( 1 + 7.41iT - 59T^{2} \) |
| 61 | \( 1 - 8.97iT - 61T^{2} \) |
| 67 | \( 1 + (8.66 + 8.66i)T + 67iT^{2} \) |
| 71 | \( 1 - 7.37iT - 71T^{2} \) |
| 73 | \( 1 + (-1.83 - 1.83i)T + 73iT^{2} \) |
| 79 | \( 1 + 8.28iT - 79T^{2} \) |
| 83 | \( 1 + (-5.27 - 5.27i)T + 83iT^{2} \) |
| 89 | \( 1 - 11.5T + 89T^{2} \) |
| 97 | \( 1 + (2.79 - 2.79i)T - 97iT^{2} \) |
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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
−14.25450614861670820298581599549, −12.84467456322552778430566841996, −12.13191435572605716896383005152, −10.44933274413554656061084476445, −9.388231513153627403344184152988, −8.949391690741773214327471578705, −6.86792575835271986540020047802, −5.95393182243141286790749156923, −5.16945843054804560102665793857, −3.46944298377312440836996918197,
1.22241965265579175325311963162, 3.10269244950074462248904383062, 5.05801583064239343517014288521, 6.19387156193611286087044171318, 7.46352304034550533453790222394, 9.309348408774495610188651341718, 10.02507606207812451567283327223, 11.11366399846570259697157666470, 12.13739212049086769910429785757, 13.11116443352523575661527797859