L(s) = 1 | + (−0.607 − 2.26i)2-s + (−2.39 − 1.38i)3-s + (−3.03 + 1.75i)4-s + (1.12 + 1.12i)5-s + (−1.67 + 6.25i)6-s + (−0.437 − 2.60i)7-s + (2.50 + 2.50i)8-s + (2.30 + 4.00i)9-s + (1.87 − 3.23i)10-s + (−3.03 + 0.813i)11-s + 9.68·12-s + (−1.04 − 3.44i)13-s + (−5.64 + 2.57i)14-s + (−1.13 − 4.24i)15-s + (0.649 − 1.12i)16-s + (0.320 + 0.555i)17-s + ⋯ |
L(s) = 1 | + (−0.429 − 1.60i)2-s + (−1.38 − 0.796i)3-s + (−1.51 + 0.877i)4-s + (0.503 + 0.503i)5-s + (−0.684 + 2.55i)6-s + (−0.165 − 0.986i)7-s + (0.886 + 0.886i)8-s + (0.769 + 1.33i)9-s + (0.591 − 1.02i)10-s + (−0.914 + 0.245i)11-s + 2.79·12-s + (−0.290 − 0.956i)13-s + (−1.51 + 0.689i)14-s + (−0.293 − 1.09i)15-s + (0.162 − 0.281i)16-s + (0.0778 + 0.134i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 - 0.584i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.811 - 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.128482 + 0.398018i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.128482 + 0.398018i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.437 + 2.60i)T \) |
| 13 | \( 1 + (1.04 + 3.44i)T \) |
good | 2 | \( 1 + (0.607 + 2.26i)T + (-1.73 + i)T^{2} \) |
| 3 | \( 1 + (2.39 + 1.38i)T + (1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.12 - 1.12i)T + 5iT^{2} \) |
| 11 | \( 1 + (3.03 - 0.813i)T + (9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (-0.320 - 0.555i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.04 + 7.61i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (0.126 + 0.0730i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.49 + 2.58i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.73 - 4.73i)T + 31iT^{2} \) |
| 37 | \( 1 + (-3.75 + 1.00i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-5.60 + 1.50i)T + (35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-2.42 + 1.40i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (2.22 - 2.22i)T - 47iT^{2} \) |
| 53 | \( 1 + 7.32T + 53T^{2} \) |
| 59 | \( 1 + (4.00 + 1.07i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (3.90 - 2.25i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.366 + 1.36i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-13.8 - 3.70i)T + (61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (-4.99 + 4.99i)T - 73iT^{2} \) |
| 79 | \( 1 + 0.632T + 79T^{2} \) |
| 83 | \( 1 + (-1.07 - 1.07i)T + 83iT^{2} \) |
| 89 | \( 1 + (-3.51 - 13.1i)T + (-77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-0.0487 + 0.181i)T + (-84.0 - 48.5i)T^{2} \) |
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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
−13.01157691192928602539784735140, −12.27855546702956916713322092552, −11.04555539496183665756004716213, −10.64144123206956189201734058192, −9.739427806516304449452815376376, −7.74191204649014390845982673392, −6.49426244525821635242296402929, −4.86664375032129609007297281374, −2.69832636322815728440675340999, −0.69568648588983374142603962967,
4.79806788931418410500684654594, 5.64207069928988648628146828255, 6.27074715885515503981777935475, 7.944576684215764238269571105055, 9.288940436957970840419645795389, 9.959023581913590165132031797090, 11.48172574909536359750226737039, 12.58395748255565605825914922602, 14.08030119155179998884426395881, 15.22166557708707324083170226781