| L(s) = 1 | + (−1.00 + 0.994i)2-s + (0.187 + 2.14i)3-s + (0.0230 − 1.99i)4-s + (−1.45 + 3.11i)5-s + (−2.31 − 1.96i)6-s + (0.904 + 1.56i)7-s + (1.96 + 2.03i)8-s + (−1.60 + 0.282i)9-s + (−1.63 − 4.58i)10-s + (−0.0590 + 0.0158i)11-s + (4.28 − 0.325i)12-s + (−0.0802 + 0.917i)13-s + (−2.46 − 0.676i)14-s + (−6.95 − 2.53i)15-s + (−3.99 − 0.0923i)16-s + (0.132 − 0.751i)17-s + ⋯ |
| L(s) = 1 | + (−0.711 + 0.703i)2-s + (0.108 + 1.23i)3-s + (0.0115 − 0.999i)4-s + (−0.650 + 1.39i)5-s + (−0.946 − 0.803i)6-s + (0.341 + 0.592i)7-s + (0.694 + 0.719i)8-s + (−0.533 + 0.0940i)9-s + (−0.517 − 1.44i)10-s + (−0.0178 + 0.00477i)11-s + (1.23 − 0.0939i)12-s + (−0.0222 + 0.254i)13-s + (−0.659 − 0.180i)14-s + (−1.79 − 0.653i)15-s + (−0.999 − 0.0230i)16-s + (0.0321 − 0.182i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 + 0.268i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.963 + 0.268i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.106373 - 0.778698i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.106373 - 0.778698i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.00 - 0.994i)T \) |
| 19 | \( 1 + (1.18 + 4.19i)T \) |
| good | 3 | \( 1 + (-0.187 - 2.14i)T + (-2.95 + 0.520i)T^{2} \) |
| 5 | \( 1 + (1.45 - 3.11i)T + (-3.21 - 3.83i)T^{2} \) |
| 7 | \( 1 + (-0.904 - 1.56i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.0590 - 0.0158i)T + (9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (0.0802 - 0.917i)T + (-12.8 - 2.25i)T^{2} \) |
| 17 | \( 1 + (-0.132 + 0.751i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-2.43 - 0.887i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-2.36 - 1.65i)T + (9.91 + 27.2i)T^{2} \) |
| 31 | \( 1 + (5.35 + 9.27i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.90 - 6.90i)T + 37iT^{2} \) |
| 41 | \( 1 + (-0.428 - 0.359i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (6.34 + 2.95i)T + (27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (7.04 - 1.24i)T + (44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-4.15 - 8.91i)T + (-34.0 + 40.6i)T^{2} \) |
| 59 | \( 1 + (-3.26 - 4.66i)T + (-20.1 + 55.4i)T^{2} \) |
| 61 | \( 1 + (-4.02 - 8.64i)T + (-39.2 + 46.7i)T^{2} \) |
| 67 | \( 1 + (-2.12 + 3.02i)T + (-22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (-4.42 - 12.1i)T + (-54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-0.0412 + 0.0491i)T + (-12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-8.32 - 6.98i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (0.133 + 0.0358i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (-2.03 + 1.70i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-17.9 - 3.17i)T + (91.1 + 33.1i)T^{2} \) |
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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
−11.53948386428038702306216698020, −11.11716707570994637735464468249, −10.23092218650664975860716793810, −9.450075789159203838706612519850, −8.534617634247100115485003142804, −7.42786154712463090359850367653, −6.57305955713692347599361253429, −5.27390345563600971244807051820, −4.14420826858637814047997055375, −2.67988526307726704234227021028,
0.75591418087418445068107308667, 1.79844747586089585616721510824, 3.69479938689649063709382288365, 4.89945196583575279507915019602, 6.71511312497834294083955410906, 7.82920598232213816240825312668, 8.150409122870584217732994231781, 9.117896486193480270683684373986, 10.34668794004583878100344376425, 11.39904602848904267977024617155
