| L(s) = 1 | + (1.73 + i)2-s + (2.86 + 4.33i)3-s + (1.99 + 3.46i)4-s + (−4.49 − 7.78i)5-s + (0.626 + 10.3i)6-s + (−3.72 + 18.1i)7-s + 7.99i·8-s + (−10.5 + 24.8i)9-s − 17.9i·10-s + (57.4 + 33.1i)11-s + (−9.28 + 18.5i)12-s + (−42.3 + 24.4i)13-s + (−24.5 + 27.6i)14-s + (20.8 − 41.7i)15-s + (−8 + 13.8i)16-s + 18.7·17-s + ⋯ |
| L(s) = 1 | + (0.612 + 0.353i)2-s + (0.551 + 0.834i)3-s + (0.249 + 0.433i)4-s + (−0.401 − 0.696i)5-s + (0.0426 + 0.705i)6-s + (−0.201 + 0.979i)7-s + 0.353i·8-s + (−0.392 + 0.919i)9-s − 0.568i·10-s + (1.57 + 0.909i)11-s + (−0.223 + 0.447i)12-s + (−0.904 + 0.521i)13-s + (−0.469 + 0.528i)14-s + (0.359 − 0.719i)15-s + (−0.125 + 0.216i)16-s + 0.267·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.263 - 0.964i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.263 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.49071 + 1.95328i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.49071 + 1.95328i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.73 - i)T \) |
| 3 | \( 1 + (-2.86 - 4.33i)T \) |
| 7 | \( 1 + (3.72 - 18.1i)T \) |
| good | 5 | \( 1 + (4.49 + 7.78i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-57.4 - 33.1i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (42.3 - 24.4i)T + (1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 - 18.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 109. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (30.4 - 17.5i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (53.0 + 30.6i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-289. + 167. i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 - 251.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-134. - 232. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-69.7 + 120. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (57.9 - 100. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + 438. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (-113. - 195. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (469. + 271. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-307. - 531. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 116. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 352. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + (-390. + 677. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (122. - 212. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 - 1.64e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (969. + 559. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−13.27569224038413581924548907824, −12.10843343515858669419372971759, −11.56910198958744055877739961824, −9.637113853160805798732439361719, −9.112127653042169322834846166190, −7.918510940035391629391361461839, −6.46403082062464946189307386390, −4.88148820661127886767168411673, −4.19588074677416513508220348336, −2.49396028214183040717015558674,
1.09672301780741750157161373193, 3.05630944735957273789143433481, 3.95803565310604661516442046625, 6.11749389296491436640769998706, 7.01606043160474936483212546086, 8.036259823936868093869347458860, 9.561886283590993446808658253037, 10.73410839388590962550336213125, 11.82621158404944195472996073877, 12.55223358282887847721650628952
