| L(s) = 1 | + (0.263 + 1.38i)2-s + (2.73 + 1.27i)3-s + (−1.86 + 0.732i)4-s + (1.07 + 1.53i)5-s + (−1.04 + 4.13i)6-s + (−1.72 − 2.99i)7-s + (−1.50 − 2.39i)8-s + (3.91 + 4.66i)9-s + (−1.84 + 1.89i)10-s + (3.78 − 1.01i)11-s + (−6.01 − 0.368i)12-s + (−4.68 + 2.18i)13-s + (3.69 − 3.18i)14-s + (0.979 + 5.55i)15-s + (2.92 − 2.72i)16-s + (−0.0194 − 0.0163i)17-s + ⋯ |
| L(s) = 1 | + (0.186 + 0.982i)2-s + (1.57 + 0.735i)3-s + (−0.930 + 0.366i)4-s + (0.479 + 0.685i)5-s + (−0.428 + 1.68i)6-s + (−0.652 − 1.13i)7-s + (−0.533 − 0.845i)8-s + (1.30 + 1.55i)9-s + (−0.583 + 0.599i)10-s + (1.14 − 0.306i)11-s + (−1.73 − 0.106i)12-s + (−1.29 + 0.605i)13-s + (0.988 − 0.852i)14-s + (0.252 + 1.43i)15-s + (0.731 − 0.681i)16-s + (−0.00472 − 0.00396i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.398 - 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.398 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.15198 + 1.75552i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15198 + 1.75552i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.263 - 1.38i)T \) |
| 19 | \( 1 + (2.67 + 3.43i)T \) |
| good | 3 | \( 1 + (-2.73 - 1.27i)T + (1.92 + 2.29i)T^{2} \) |
| 5 | \( 1 + (-1.07 - 1.53i)T + (-1.71 + 4.69i)T^{2} \) |
| 7 | \( 1 + (1.72 + 2.99i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.78 + 1.01i)T + (9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (4.68 - 2.18i)T + (8.35 - 9.95i)T^{2} \) |
| 17 | \( 1 + (0.0194 + 0.0163i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (0.130 + 0.742i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.595 + 6.80i)T + (-28.5 - 5.03i)T^{2} \) |
| 31 | \( 1 + (-4.03 - 6.99i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.91 + 3.91i)T + 37iT^{2} \) |
| 41 | \( 1 + (4.18 - 1.52i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (0.482 - 0.337i)T + (14.7 - 40.4i)T^{2} \) |
| 47 | \( 1 + (-3.71 - 4.43i)T + (-8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (4.54 - 6.49i)T + (-18.1 - 49.8i)T^{2} \) |
| 59 | \( 1 + (0.292 - 0.0256i)T + (58.1 - 10.2i)T^{2} \) |
| 61 | \( 1 + (-3.07 + 4.39i)T + (-20.8 - 57.3i)T^{2} \) |
| 67 | \( 1 + (6.62 + 0.579i)T + (65.9 + 11.6i)T^{2} \) |
| 71 | \( 1 + (-0.833 - 0.146i)T + (66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-5.49 - 15.0i)T + (-55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-11.7 + 4.27i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-16.9 - 4.54i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (13.1 + 4.77i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (2.61 - 3.11i)T + (-16.8 - 95.5i)T^{2} \) |
| 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
−12.35339555935257538532386593285, −10.57005259424867783008267101249, −9.743016096796116543870973990829, −9.248128202082697792277226406216, −8.212199490833730767142559442610, −7.07111154507949729591651368389, −6.56466556745645339103387850244, −4.60265417060046935705279761086, −3.85974456922290152907568575036, −2.73462384957420416185091388941,
1.65412350484440432042279672792, 2.58345201972462024974249410419, 3.66033361538501984132993281849, 5.17427284333610532572617257615, 6.52770624995322493679637285406, 7.995063585753671504690109841378, 8.941105575415515371607142198647, 9.335865687793265804750912886249, 10.11255704298990363549037163133, 12.07748643848846228865025718169
