L(s) = 1 | + (2.44 + 4.58i)3-s − 17.2·5-s − 0.269i·7-s + (−15 + 22.4i)9-s − 41.1i·11-s − 64.6i·13-s + (−42.3 − 79.1i)15-s + 63.4i·17-s + 154.·19-s + (1.23 − 0.660i)21-s − 43.3·23-s + 173.·25-s + (−139. − 13.7i)27-s + 240.·29-s + 79.1i·31-s + ⋯ |
L(s) = 1 | + (0.471 + 0.881i)3-s − 1.54·5-s − 0.0145i·7-s + (−0.555 + 0.831i)9-s − 1.12i·11-s − 1.37i·13-s + (−0.728 − 1.36i)15-s + 0.904i·17-s + 1.85·19-s + (0.0128 − 0.00686i)21-s − 0.392·23-s + 1.38·25-s + (−0.995 − 0.0979i)27-s + 1.54·29-s + 0.458i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.396823894\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.396823894\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.44 - 4.58i)T \) |
good | 5 | \( 1 + 17.2T + 125T^{2} \) |
| 7 | \( 1 + 0.269iT - 343T^{2} \) |
| 11 | \( 1 + 41.1iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 64.6iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 63.4iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 154.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 43.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 240.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 79.1iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 132. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 101. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 120.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 293.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 6.17T + 1.48e5T^{2} \) |
| 59 | \( 1 - 45.8iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 651. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 685.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 836.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 285.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 940. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 1.01e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 432. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 1.00e3T + 9.12e5T^{2} \) |
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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
−10.82679203300196649222654940875, −10.12358075381004856130152448774, −8.817345332214858537716387811299, −8.133028867931426739151730572066, −7.54738059446851909058873940699, −5.84642339151171680112903754472, −4.79484162651440739982909314960, −3.55989685177603178916149582489, −3.13130538175012750713476397562, −0.57520644305495793112511196890,
1.03882729035575698699368683502, 2.62441086887628283933217215185, 3.84408653621172773166330098432, 4.87194396603771040040501512740, 6.63791160331797031286201283029, 7.32888658702023774116303303448, 7.911509053340047543974353039325, 8.992727335858666058111056697521, 9.840547854025540926293643490704, 11.52431814925013136849235577487