L(s) = 1 | + (2.57 + 0.690i)3-s + (2.23 − 0.0143i)5-s + (−1.08 + 1.87i)7-s + (3.56 + 2.06i)9-s + (−5.13 − 1.37i)11-s + (−2.72 − 2.35i)13-s + (5.77 + 1.50i)15-s + (0.288 + 1.07i)17-s + (−1.69 − 6.33i)19-s + (−4.08 + 4.08i)21-s + (−0.192 + 0.718i)23-s + (4.99 − 0.0642i)25-s + (2.11 + 2.11i)27-s + (0.0866 − 0.0500i)29-s + (3.90 + 3.90i)31-s + ⋯ |
L(s) = 1 | + (1.48 + 0.398i)3-s + (0.999 − 0.00642i)5-s + (−0.408 + 0.708i)7-s + (1.18 + 0.686i)9-s + (−1.54 − 0.414i)11-s + (−0.757 − 0.653i)13-s + (1.49 + 0.389i)15-s + (0.0699 + 0.260i)17-s + (−0.389 − 1.45i)19-s + (−0.890 + 0.890i)21-s + (−0.0401 + 0.149i)23-s + (0.999 − 0.0128i)25-s + (0.406 + 0.406i)27-s + (0.0160 − 0.00929i)29-s + (0.700 + 0.700i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 - 0.398i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.917 - 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.95986 + 0.407703i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.95986 + 0.407703i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2.23 + 0.0143i)T \) |
| 13 | \( 1 + (2.72 + 2.35i)T \) |
good | 3 | \( 1 + (-2.57 - 0.690i)T + (2.59 + 1.5i)T^{2} \) |
| 7 | \( 1 + (1.08 - 1.87i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (5.13 + 1.37i)T + (9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (-0.288 - 1.07i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (1.69 + 6.33i)T + (-16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (0.192 - 0.718i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (-0.0866 + 0.0500i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.90 - 3.90i)T + 31iT^{2} \) |
| 37 | \( 1 + (-3.41 - 5.91i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.36 + 5.10i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-3.57 + 0.959i)T + (37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 - 2.04T + 47T^{2} \) |
| 53 | \( 1 + (-8.28 + 8.28i)T - 53iT^{2} \) |
| 59 | \( 1 + (12.1 - 3.25i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (4.66 - 8.08i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (9.54 - 5.51i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (9.19 - 2.46i)T + (61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + 10.7iT - 73T^{2} \) |
| 79 | \( 1 - 15.3iT - 79T^{2} \) |
| 83 | \( 1 - 0.473T + 83T^{2} \) |
| 89 | \( 1 + (-0.560 + 2.09i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (-10.9 - 6.34i)T + (48.5 + 84.0i)T^{2} \) |
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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
−12.38389932000541900712089273676, −10.65755843512845852794070423231, −10.00907293048234709215844620813, −9.096645162915552187894839782794, −8.452828391660039298287564338038, −7.36125638987316305366170685829, −5.85851583118023189797611113619, −4.80534764175360936488619663203, −2.87291599146952085741073903089, −2.54340330828619056062641597553,
2.00012914493419938429366168946, 2.91397709444772044590990889825, 4.44676230054554322044166428378, 5.99046711432955020574987307254, 7.31904509461510067414762456846, 7.896738088320110446607027698599, 9.128268014998640087002368649631, 9.898964048803078691112068024868, 10.54581957629565257193493027215, 12.37383283230080853794129875190