| L(s) = 1 | + (−1.35 − 0.398i)2-s + (1.68 + 1.08i)4-s + (3.34 + 0.480i)5-s + (−3.29 + 1.50i)7-s + (−1.85 − 2.13i)8-s + (−4.34 − 1.98i)10-s + (0.832 + 2.83i)11-s + (−3.91 + 8.58i)13-s + (5.07 − 0.729i)14-s + (1.66 + 3.63i)16-s + (15.1 + 23.6i)17-s + (5.89 − 9.17i)19-s + (5.10 + 4.42i)20-s − 4.17i·22-s + (−19.6 − 12.0i)23-s + ⋯ |
| L(s) = 1 | + (−0.678 − 0.199i)2-s + (0.420 + 0.270i)4-s + (0.668 + 0.0960i)5-s + (−0.471 + 0.215i)7-s + (−0.231 − 0.267i)8-s + (−0.434 − 0.198i)10-s + (0.0756 + 0.257i)11-s + (−0.301 + 0.660i)13-s + (0.362 − 0.0521i)14-s + (0.103 + 0.227i)16-s + (0.892 + 1.38i)17-s + (0.310 − 0.483i)19-s + (0.255 + 0.221i)20-s − 0.189i·22-s + (−0.852 − 0.522i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.244 - 0.969i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.244 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.843961 + 0.657530i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.843961 + 0.657530i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.35 + 0.398i)T \) |
| 3 | \( 1 \) |
| 23 | \( 1 + (19.6 + 12.0i)T \) |
| good | 5 | \( 1 + (-3.34 - 0.480i)T + (23.9 + 7.04i)T^{2} \) |
| 7 | \( 1 + (3.29 - 1.50i)T + (32.0 - 37.0i)T^{2} \) |
| 11 | \( 1 + (-0.832 - 2.83i)T + (-101. + 65.4i)T^{2} \) |
| 13 | \( 1 + (3.91 - 8.58i)T + (-110. - 127. i)T^{2} \) |
| 17 | \( 1 + (-15.1 - 23.6i)T + (-120. + 262. i)T^{2} \) |
| 19 | \( 1 + (-5.89 + 9.17i)T + (-149. - 328. i)T^{2} \) |
| 29 | \( 1 + (26.1 - 16.7i)T + (349. - 765. i)T^{2} \) |
| 31 | \( 1 + (-18.1 - 20.9i)T + (-136. + 951. i)T^{2} \) |
| 37 | \( 1 + (-11.0 + 1.58i)T + (1.31e3 - 385. i)T^{2} \) |
| 41 | \( 1 + (9.53 - 66.2i)T + (-1.61e3 - 473. i)T^{2} \) |
| 43 | \( 1 + (-35.0 - 30.3i)T + (263. + 1.83e3i)T^{2} \) |
| 47 | \( 1 - 54.3T + 2.20e3T^{2} \) |
| 53 | \( 1 + (-64.7 + 29.5i)T + (1.83e3 - 2.12e3i)T^{2} \) |
| 59 | \( 1 + (26.5 - 58.0i)T + (-2.27e3 - 2.63e3i)T^{2} \) |
| 61 | \( 1 + (65.2 - 56.4i)T + (529. - 3.68e3i)T^{2} \) |
| 67 | \( 1 + (0.347 - 1.18i)T + (-3.77e3 - 2.42e3i)T^{2} \) |
| 71 | \( 1 + (-66.9 - 19.6i)T + (4.24e3 + 2.72e3i)T^{2} \) |
| 73 | \( 1 + (-15.1 - 9.72i)T + (2.21e3 + 4.84e3i)T^{2} \) |
| 79 | \( 1 + (-94.7 - 43.2i)T + (4.08e3 + 4.71e3i)T^{2} \) |
| 83 | \( 1 + (84.2 - 12.1i)T + (6.60e3 - 1.94e3i)T^{2} \) |
| 89 | \( 1 + (21.6 + 18.7i)T + (1.12e3 + 7.84e3i)T^{2} \) |
| 97 | \( 1 + (151. + 21.8i)T + (9.02e3 + 2.65e3i)T^{2} \) |
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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
−11.01170614437947561235099968045, −10.05867363727851124469613820483, −9.577042289202560810657030799321, −8.600783601880317935535291686995, −7.59169562072794684921474624327, −6.49814066320893262719032709714, −5.72946936678365371388126293928, −4.14777706623150174638852814215, −2.74432731813660326180200495287, −1.52073750743264065170060888946,
0.58110470539370562490357348439, 2.24023425591934905931813273240, 3.61112692772766057483107381410, 5.38330523949180374401661659531, 5.99895231720368082720471447451, 7.30211104301451078611297860720, 7.924967427666053320054734216043, 9.290385478944806368186096308843, 9.738963848900137112751396124973, 10.51347903643092310611591583542
