| L(s) = 1 | + (1.36 + 0.366i)2-s + (0.714 + 0.232i)3-s + (1.73 + 1.00i)4-s + (0.654 + 2.13i)5-s + (0.890 + 0.578i)6-s + (0.664 + 0.664i)7-s + (1.99 + 2.00i)8-s + (−1.97 − 1.43i)9-s + (0.110 + 3.16i)10-s + (−1.89 − 0.300i)11-s + (1.00 + 1.11i)12-s + (−2.19 − 1.59i)13-s + (0.664 + 1.15i)14-s + (−0.0284 + 1.67i)15-s + (1.99 + 3.46i)16-s + (1.25 − 2.46i)17-s + ⋯ |
| L(s) = 1 | + (0.965 + 0.259i)2-s + (0.412 + 0.133i)3-s + (0.865 + 0.500i)4-s + (0.292 + 0.956i)5-s + (0.363 + 0.236i)6-s + (0.251 + 0.251i)7-s + (0.706 + 0.708i)8-s + (−0.656 − 0.477i)9-s + (0.0349 + 0.999i)10-s + (−0.572 − 0.0906i)11-s + (0.289 + 0.322i)12-s + (−0.608 − 0.442i)13-s + (0.177 + 0.307i)14-s + (−0.00735 + 0.433i)15-s + (0.498 + 0.866i)16-s + (0.304 − 0.597i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.534 - 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.534 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.39498 + 1.31829i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.39498 + 1.31829i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.36 - 0.366i)T \) |
| 5 | \( 1 + (-0.654 - 2.13i)T \) |
| good | 3 | \( 1 + (-0.714 - 0.232i)T + (2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 + (-0.664 - 0.664i)T + 7iT^{2} \) |
| 11 | \( 1 + (1.89 + 0.300i)T + (10.4 + 3.39i)T^{2} \) |
| 13 | \( 1 + (2.19 + 1.59i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-1.25 + 2.46i)T + (-9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (-0.627 + 1.23i)T + (-11.1 - 15.3i)T^{2} \) |
| 23 | \( 1 + (-0.899 + 5.67i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (-1.42 - 2.80i)T + (-17.0 + 23.4i)T^{2} \) |
| 31 | \( 1 + (-8.03 + 2.61i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-1.63 - 1.18i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (1.21 - 1.67i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 5.07T + 43T^{2} \) |
| 47 | \( 1 + (-3.59 - 7.05i)T + (-27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (12.2 + 3.99i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (3.22 - 0.511i)T + (56.1 - 18.2i)T^{2} \) |
| 61 | \( 1 + (3.20 + 0.507i)T + (58.0 + 18.8i)T^{2} \) |
| 67 | \( 1 + (-3.69 - 11.3i)T + (-54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (-0.485 + 1.49i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (15.8 + 2.51i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (-3.95 + 12.1i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (7.98 - 2.59i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (11.2 - 8.20i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (5.10 - 2.59i)T + (57.0 - 78.4i)T^{2} \) |
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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
−11.52850539010657365434982565146, −10.68788692113292783469056711251, −9.738901650791989917053810991140, −8.422003543295839891098527851176, −7.56445910400051071025006741034, −6.54026704881665722009512628806, −5.67712258617486863039990714387, −4.58187534215469046297635442691, −3.02979769175300416858624470982, −2.61643700341193244438532435305,
1.62253379335668762621334102345, 2.84640448092108809667948209314, 4.28865515724027068068887279761, 5.18181919848137331377231214706, 6.00690627630255348549883665654, 7.46636147793215572223682767884, 8.203319865067864833220114825041, 9.427693451893849025097957246485, 10.32423300627224309781107801650, 11.34304300114131379130158109181
