| L(s) = 1 | + (1.28 − 0.587i)2-s + (−0.540 − 1.84i)3-s + (1.30 − 1.51i)4-s + (−3.54 − 3.52i)5-s + (−1.77 − 2.05i)6-s + (0.0656 + 0.456i)7-s + (0.796 − 2.71i)8-s + (4.47 − 2.87i)9-s + (−6.63 − 2.44i)10-s + (0.696 + 0.318i)11-s + (−3.49 − 1.59i)12-s + (−20.8 − 2.99i)13-s + (0.352 + 0.549i)14-s + (−4.56 + 8.43i)15-s + (−0.569 − 3.95i)16-s + (2.37 + 2.74i)17-s + ⋯ |
| L(s) = 1 | + (0.643 − 0.293i)2-s + (−0.180 − 0.613i)3-s + (0.327 − 0.377i)4-s + (−0.709 − 0.704i)5-s + (−0.296 − 0.341i)6-s + (0.00938 + 0.0652i)7-s + (0.0996 − 0.339i)8-s + (0.496 − 0.319i)9-s + (−0.663 − 0.244i)10-s + (0.0633 + 0.0289i)11-s + (−0.290 − 0.132i)12-s + (−1.60 − 0.230i)13-s + (0.0252 + 0.0392i)14-s + (−0.304 + 0.562i)15-s + (−0.0355 − 0.247i)16-s + (0.139 + 0.161i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.800 + 0.598i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.800 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.499986 - 1.50316i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.499986 - 1.50316i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.28 + 0.587i)T \) |
| 5 | \( 1 + (3.54 + 3.52i)T \) |
| 23 | \( 1 + (-9.91 + 20.7i)T \) |
| good | 3 | \( 1 + (0.540 + 1.84i)T + (-7.57 + 4.86i)T^{2} \) |
| 7 | \( 1 + (-0.0656 - 0.456i)T + (-47.0 + 13.8i)T^{2} \) |
| 11 | \( 1 + (-0.696 - 0.318i)T + (79.2 + 91.4i)T^{2} \) |
| 13 | \( 1 + (20.8 + 2.99i)T + (162. + 47.6i)T^{2} \) |
| 17 | \( 1 + (-2.37 - 2.74i)T + (-41.1 + 286. i)T^{2} \) |
| 19 | \( 1 + (22.1 + 19.1i)T + (51.3 + 357. i)T^{2} \) |
| 29 | \( 1 + (-26.9 - 31.1i)T + (-119. + 832. i)T^{2} \) |
| 31 | \( 1 + (-35.8 - 10.5i)T + (808. + 519. i)T^{2} \) |
| 37 | \( 1 + (-57.6 + 37.0i)T + (568. - 1.24e3i)T^{2} \) |
| 41 | \( 1 + (30.9 + 19.9i)T + (698. + 1.52e3i)T^{2} \) |
| 43 | \( 1 + (-31.1 + 9.14i)T + (1.55e3 - 9.99e2i)T^{2} \) |
| 47 | \( 1 - 60.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (11.2 + 78.3i)T + (-2.69e3 + 791. i)T^{2} \) |
| 59 | \( 1 + (8.45 - 58.7i)T + (-3.33e3 - 980. i)T^{2} \) |
| 61 | \( 1 + (-30.8 + 105. i)T + (-3.13e3 - 2.01e3i)T^{2} \) |
| 67 | \( 1 + (-45.4 - 99.4i)T + (-2.93e3 + 3.39e3i)T^{2} \) |
| 71 | \( 1 + (33.8 + 74.0i)T + (-3.30e3 + 3.80e3i)T^{2} \) |
| 73 | \( 1 + (-72.5 - 62.8i)T + (758. + 5.27e3i)T^{2} \) |
| 79 | \( 1 + (21.5 + 3.09i)T + (5.98e3 + 1.75e3i)T^{2} \) |
| 83 | \( 1 + (75.4 - 48.4i)T + (2.86e3 - 6.26e3i)T^{2} \) |
| 89 | \( 1 + (-5.79 - 19.7i)T + (-6.66e3 + 4.28e3i)T^{2} \) |
| 97 | \( 1 + (-58.1 - 37.3i)T + (3.90e3 + 8.55e3i)T^{2} \) |
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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
−11.98766529643672956100777363729, −10.89960107294693540270101128783, −9.755498467219426691474698001513, −8.561293967067496006459600972177, −7.36253580979870374529377414957, −6.56072397570142581696792003836, −5.02447439044009472284994393843, −4.24218012523968573651498984479, −2.51798369148252717638704478958, −0.71322133917808364438960200307,
2.60308369791842350244772651248, 4.07051611844256416131217120450, 4.76561784409211835611226306620, 6.22907090372843876527214714989, 7.31944886776235187557611095886, 8.071023409593910592755825508888, 9.775491158206661311487765054424, 10.43302053765855377756657981359, 11.57140005652070139030178262248, 12.20694347426497524103806068332
