| L(s) = 1 | + (1.39 + 0.245i)2-s + (−3.75 + 4.47i)3-s + (1.87 + 0.684i)4-s + (5.30 − 1.92i)5-s + (−6.32 + 5.31i)6-s + (−0.990 − 1.71i)7-s + (2.44 + 1.41i)8-s + (−4.36 − 24.7i)9-s + (7.85 − 1.38i)10-s + (1.21 − 2.09i)11-s + (−10.1 + 5.84i)12-s + (1.83 + 2.19i)13-s + (−0.957 − 2.63i)14-s + (−11.2 + 30.9i)15-s + (3.06 + 2.57i)16-s + (−2.41 + 13.7i)17-s + ⋯ |
| L(s) = 1 | + (0.696 + 0.122i)2-s + (−1.25 + 1.49i)3-s + (0.469 + 0.171i)4-s + (1.06 − 0.385i)5-s + (−1.05 + 0.885i)6-s + (−0.141 − 0.245i)7-s + (0.306 + 0.176i)8-s + (−0.484 − 2.75i)9-s + (0.785 − 0.138i)10-s + (0.110 − 0.190i)11-s + (−0.843 + 0.486i)12-s + (0.141 + 0.168i)13-s + (−0.0684 − 0.188i)14-s + (−0.751 + 2.06i)15-s + (0.191 + 0.160i)16-s + (−0.142 + 0.805i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.468 - 0.883i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.468 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.02409 + 0.616314i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02409 + 0.616314i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.39 - 0.245i)T \) |
| 19 | \( 1 + (-8.50 + 16.9i)T \) |
| good | 3 | \( 1 + (3.75 - 4.47i)T + (-1.56 - 8.86i)T^{2} \) |
| 5 | \( 1 + (-5.30 + 1.92i)T + (19.1 - 16.0i)T^{2} \) |
| 7 | \( 1 + (0.990 + 1.71i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-1.21 + 2.09i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-1.83 - 2.19i)T + (-29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (2.41 - 13.7i)T + (-271. - 98.8i)T^{2} \) |
| 23 | \( 1 + (33.6 + 12.2i)T + (405. + 340. i)T^{2} \) |
| 29 | \( 1 + (28.0 - 4.94i)T + (790. - 287. i)T^{2} \) |
| 31 | \( 1 + (1.64 - 0.949i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 15.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (14.3 - 17.1i)T + (-291. - 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-30.7 + 11.1i)T + (1.41e3 - 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-8.75 - 49.6i)T + (-2.07e3 + 755. i)T^{2} \) |
| 53 | \( 1 + (9.90 - 27.2i)T + (-2.15e3 - 1.80e3i)T^{2} \) |
| 59 | \( 1 + (54.0 + 9.53i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-53.0 - 19.3i)T + (2.85e3 + 2.39e3i)T^{2} \) |
| 67 | \( 1 + (-42.9 + 7.57i)T + (4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (7.09 + 19.4i)T + (-3.86e3 + 3.24e3i)T^{2} \) |
| 73 | \( 1 + (-4.76 - 3.99i)T + (925. + 5.24e3i)T^{2} \) |
| 79 | \( 1 + (44.0 - 52.4i)T + (-1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-37.7 - 65.3i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-48.5 - 57.8i)T + (-1.37e3 + 7.80e3i)T^{2} \) |
| 97 | \( 1 + (91.8 + 16.1i)T + (8.84e3 + 3.21e3i)T^{2} \) |
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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
−16.31597539525806473893378493828, −15.33568239036235216897757990415, −14.05928119771511215344116239966, −12.63589226829214904505985311944, −11.34426767696361672349804525240, −10.30439173115157105951413924801, −9.235766500435758713321819706146, −6.30583539869949903715199023694, −5.40854596780748665152299365325, −4.05570443369918225130600083804,
1.99744354433242544281864345662, 5.50285856592074581169923590262, 6.25059953825157187983344439591, 7.52843456678476013362684101940, 10.11899836662617502240322962920, 11.46586892014904954700519365781, 12.34315556632689000919653039701, 13.43148218335872446736022068689, 14.11687899939112673808097997521, 16.09067367162262169941279122300
