| L(s) = 1 | + (−3.23 − 2.35i)2-s + (1.31 + 12.5i)3-s + (4.94 + 15.2i)4-s + (−23.3 − 40.4i)5-s + (25.1 − 43.6i)6-s + (−78.2 − 16.6i)7-s + (19.7 − 60.8i)8-s + (82.4 − 17.5i)9-s + (−19.5 + 185. i)10-s + (53.9 − 59.9i)11-s + (−184. + 81.9i)12-s + (764. + 340. i)13-s + (214. + 237. i)14-s + (476. − 346. i)15-s + (−207. + 150. i)16-s + (108. + 120. i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 − 0.415i)2-s + (0.0844 + 0.803i)3-s + (0.154 + 0.475i)4-s + (−0.418 − 0.724i)5-s + (0.285 − 0.494i)6-s + (−0.603 − 0.128i)7-s + (0.109 − 0.336i)8-s + (0.339 − 0.0721i)9-s + (−0.0617 + 0.587i)10-s + (0.134 − 0.149i)11-s + (−0.369 + 0.164i)12-s + (1.25 + 0.558i)13-s + (0.292 + 0.324i)14-s + (0.546 − 0.397i)15-s + (−0.202 + 0.146i)16-s + (0.0911 + 0.101i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 + 0.484i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.874 + 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.21526 - 0.313974i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.21526 - 0.313974i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.23 + 2.35i)T \) |
| 31 | \( 1 + (-5.33e3 - 459. i)T \) |
| good | 3 | \( 1 + (-1.31 - 12.5i)T + (-237. + 50.5i)T^{2} \) |
| 5 | \( 1 + (23.3 + 40.4i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 7 | \( 1 + (78.2 + 16.6i)T + (1.53e4 + 6.83e3i)T^{2} \) |
| 11 | \( 1 + (-53.9 + 59.9i)T + (-1.68e4 - 1.60e5i)T^{2} \) |
| 13 | \( 1 + (-764. - 340. i)T + (2.48e5 + 2.75e5i)T^{2} \) |
| 17 | \( 1 + (-108. - 120. i)T + (-1.48e5 + 1.41e6i)T^{2} \) |
| 19 | \( 1 + (-2.25e3 + 1.00e3i)T + (1.65e6 - 1.84e6i)T^{2} \) |
| 23 | \( 1 + (-1.17e3 + 3.62e3i)T + (-5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (1.49e3 + 1.08e3i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 37 | \( 1 + (5.21e3 - 9.03e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + (95.2 - 906. i)T + (-1.13e8 - 2.40e7i)T^{2} \) |
| 43 | \( 1 + (1.98e3 - 885. i)T + (9.83e7 - 1.09e8i)T^{2} \) |
| 47 | \( 1 + (-1.58e4 + 1.15e4i)T + (7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (8.20e3 - 1.74e3i)T + (3.82e8 - 1.70e8i)T^{2} \) |
| 59 | \( 1 + (3.22e3 + 3.06e4i)T + (-6.99e8 + 1.48e8i)T^{2} \) |
| 61 | \( 1 + 9.68e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + (-2.20e3 - 3.82e3i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (-2.14e4 + 4.56e3i)T + (1.64e9 - 7.33e8i)T^{2} \) |
| 73 | \( 1 + (9.22e3 - 1.02e4i)T + (-2.16e8 - 2.06e9i)T^{2} \) |
| 79 | \( 1 + (8.65e3 + 9.60e3i)T + (-3.21e8 + 3.06e9i)T^{2} \) |
| 83 | \( 1 + (1.55e3 - 1.48e4i)T + (-3.85e9 - 8.18e8i)T^{2} \) |
| 89 | \( 1 + (-2.49e4 - 7.67e4i)T + (-4.51e9 + 3.28e9i)T^{2} \) |
| 97 | \( 1 + (-2.91e4 - 8.97e4i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
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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
−13.76171552707589747228919991444, −12.63407415437816910565871611509, −11.49610478415120711383893356485, −10.31991395487803694842636549983, −9.295859287429798437829992853994, −8.414531157169604758239237703554, −6.70913652204581827872897698244, −4.63155593011348852466679940098, −3.39633171449118211727065213616, −0.913928157416013074346976977410,
1.21833066918722027110531459231, 3.33018423256056208129671723956, 5.83870639674737193904165514862, 7.07624498503437185346758955454, 7.81315398652254941740922120899, 9.341401462385902128146824206835, 10.58550103552489102033057078057, 11.79310362042454085063241977839, 13.10504422962454254077458009487, 14.06571108133123202033209103689
