| L(s) = 1 | + (0.164 + 0.0336i)2-s + (−1.12 + 0.182i)3-s + (−1.81 − 0.772i)4-s + (−0.123 + 0.501i)5-s + (−0.191 − 0.00772i)6-s + (−2.88 + 1.37i)7-s + (−0.549 − 0.379i)8-s + (−1.61 + 0.538i)9-s + (−0.0372 + 0.0784i)10-s + (0.240 − 0.719i)11-s + (2.18 + 0.537i)12-s + (−3.58 − 0.358i)13-s + (−0.522 + 0.128i)14-s + (0.0473 − 0.586i)15-s + (2.65 + 2.76i)16-s + (0.268 + 0.565i)17-s + ⋯ |
| L(s) = 1 | + (0.116 + 0.0237i)2-s + (−0.649 + 0.105i)3-s + (−0.906 − 0.386i)4-s + (−0.0552 + 0.224i)5-s + (−0.0782 − 0.00315i)6-s + (−1.09 + 0.518i)7-s + (−0.194 − 0.134i)8-s + (−0.537 + 0.179i)9-s + (−0.0117 + 0.0248i)10-s + (0.0723 − 0.216i)11-s + (0.629 + 0.155i)12-s + (−0.995 − 0.0993i)13-s + (−0.139 + 0.0344i)14-s + (0.0122 − 0.151i)15-s + (0.663 + 0.690i)16-s + (0.0650 + 0.137i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.161i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 - 0.161i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00745458 + 0.0919399i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00745458 + 0.0919399i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (3.58 + 0.358i)T \) |
| good | 2 | \( 1 + (-0.164 - 0.0336i)T + (1.83 + 0.783i)T^{2} \) |
| 3 | \( 1 + (1.12 - 0.182i)T + (2.84 - 0.950i)T^{2} \) |
| 5 | \( 1 + (0.123 - 0.501i)T + (-4.42 - 2.32i)T^{2} \) |
| 7 | \( 1 + (2.88 - 1.37i)T + (4.42 - 5.42i)T^{2} \) |
| 11 | \( 1 + (-0.240 + 0.719i)T + (-8.79 - 6.60i)T^{2} \) |
| 17 | \( 1 + (-0.268 - 0.565i)T + (-10.7 + 13.1i)T^{2} \) |
| 19 | \( 1 + (0.570 + 0.329i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.66 + 6.34i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.52 - 7.47i)T + (-26.6 - 11.3i)T^{2} \) |
| 31 | \( 1 + (0.765 + 1.45i)T + (-17.6 + 25.5i)T^{2} \) |
| 37 | \( 1 + (-5.47 + 8.65i)T + (-15.8 - 33.4i)T^{2} \) |
| 41 | \( 1 + (-1.31 - 8.10i)T + (-38.8 + 12.9i)T^{2} \) |
| 43 | \( 1 + (8.14 - 5.14i)T + (18.4 - 38.8i)T^{2} \) |
| 47 | \( 1 + (10.3 + 1.25i)T + (45.6 + 11.2i)T^{2} \) |
| 53 | \( 1 + (0.232 - 0.336i)T + (-18.7 - 49.5i)T^{2} \) |
| 59 | \( 1 + (4.85 + 4.66i)T + (2.37 + 58.9i)T^{2} \) |
| 61 | \( 1 + (-1.41 + 0.113i)T + (60.2 - 9.78i)T^{2} \) |
| 67 | \( 1 + (-1.84 - 4.32i)T + (-46.4 + 48.3i)T^{2} \) |
| 71 | \( 1 + (6.31 - 5.15i)T + (14.2 - 69.5i)T^{2} \) |
| 73 | \( 1 + (1.09 + 1.23i)T + (-8.79 + 72.4i)T^{2} \) |
| 79 | \( 1 + (-0.740 + 6.09i)T + (-76.7 - 18.9i)T^{2} \) |
| 83 | \( 1 + (11.0 - 4.19i)T + (62.1 - 55.0i)T^{2} \) |
| 89 | \( 1 + (8.98 - 5.18i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (4.01 + 1.16i)T + (81.9 + 51.8i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.97476049627862431815476622964, −12.50407970423055231210360344362, −11.22856724154675216090908958666, −10.19394819281650188533661598508, −9.363003627930660599028162426995, −8.324029667015693396150885996391, −6.61791658309277101405614092819, −5.72293238106079440174334058238, −4.68909704964674619506030494101, −3.02291886763891055044279419828,
0.087199298685615539982989988445, 3.23764465560813193928040610102, 4.55635421670868766274473276583, 5.74115788158287987580685865453, 6.96239426832465463630614458334, 8.206744661109874459225795077247, 9.462271229484340141535331368744, 10.08254818528205320618198843880, 11.64690870576113268837867642961, 12.31905592249340136415814530125
