L(s) = 1 | + (−0.989 + 0.989i)3-s + (0.707 + 0.707i)5-s − i·7-s + 1.04i·9-s + (3.71 + 3.71i)11-s + (2.81 − 2.81i)13-s − 1.39·15-s + 0.895·17-s + (3.33 − 3.33i)19-s + (0.989 + 0.989i)21-s − 6.38i·23-s + 1.00i·25-s + (−3.99 − 3.99i)27-s + (1.18 − 1.18i)29-s + 6.25·31-s + ⋯ |
L(s) = 1 | + (−0.571 + 0.571i)3-s + (0.316 + 0.316i)5-s − 0.377i·7-s + 0.346i·9-s + (1.12 + 1.12i)11-s + (0.780 − 0.780i)13-s − 0.361·15-s + 0.217·17-s + (0.765 − 0.765i)19-s + (0.215 + 0.215i)21-s − 1.33i·23-s + 0.200i·25-s + (−0.769 − 0.769i)27-s + (0.219 − 0.219i)29-s + 1.12·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 - 0.631i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.794792037\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.794792037\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.707 - 0.707i)T \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 + (0.989 - 0.989i)T - 3iT^{2} \) |
| 11 | \( 1 + (-3.71 - 3.71i)T + 11iT^{2} \) |
| 13 | \( 1 + (-2.81 + 2.81i)T - 13iT^{2} \) |
| 17 | \( 1 - 0.895T + 17T^{2} \) |
| 19 | \( 1 + (-3.33 + 3.33i)T - 19iT^{2} \) |
| 23 | \( 1 + 6.38iT - 23T^{2} \) |
| 29 | \( 1 + (-1.18 + 1.18i)T - 29iT^{2} \) |
| 31 | \( 1 - 6.25T + 31T^{2} \) |
| 37 | \( 1 + (4.00 + 4.00i)T + 37iT^{2} \) |
| 41 | \( 1 - 2.72iT - 41T^{2} \) |
| 43 | \( 1 + (-8.61 - 8.61i)T + 43iT^{2} \) |
| 47 | \( 1 + 3.21T + 47T^{2} \) |
| 53 | \( 1 + (3.27 + 3.27i)T + 53iT^{2} \) |
| 59 | \( 1 + (7.51 + 7.51i)T + 59iT^{2} \) |
| 61 | \( 1 + (-0.859 + 0.859i)T - 61iT^{2} \) |
| 67 | \( 1 + (-7.29 + 7.29i)T - 67iT^{2} \) |
| 71 | \( 1 - 5.87iT - 71T^{2} \) |
| 73 | \( 1 - 14.4iT - 73T^{2} \) |
| 79 | \( 1 - 4.29T + 79T^{2} \) |
| 83 | \( 1 + (-10.3 + 10.3i)T - 83iT^{2} \) |
| 89 | \( 1 + 0.769iT - 89T^{2} \) |
| 97 | \( 1 - 16.7T + 97T^{2} \) |
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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
−9.400073409085242919540293933755, −8.323699883970295612275678203535, −7.53808526794800055573198023381, −6.61962047237756660150001122124, −6.07687059179365906134872889658, −4.96542579260559230352381834506, −4.47245528046319710866353529493, −3.45916459446844943198684229902, −2.30644565495871769268009071889, −0.965518660887989484589420509399,
0.978192660353779711581100735246, 1.63644485970848651950111190050, 3.27412786267934133206735245333, 3.93232568751099801530797756927, 5.25252038950820742365787311810, 6.01388069478784334383695117090, 6.35413340997950329520887085104, 7.27525473281519846777199347098, 8.256271132056861253390344486458, 9.090230238466406224242455335649