| L(s) = 1 | + (0.932 − 1.06i)2-s + (1.44 − 0.959i)3-s + (−0.261 − 1.98i)4-s + (1.84 + 1.25i)5-s + (0.323 − 2.42i)6-s − 2.80·7-s + (−2.35 − 1.57i)8-s + (1.15 − 2.76i)9-s + (3.06 − 0.792i)10-s + (−1.50 − 1.08i)11-s + (−2.27 − 2.60i)12-s + (2.01 + 2.76i)13-s + (−2.61 + 2.98i)14-s + (3.87 + 0.0397i)15-s + (−3.86 + 1.03i)16-s + (−0.345 + 1.06i)17-s + ⋯ |
| L(s) = 1 | + (0.659 − 0.751i)2-s + (0.832 − 0.554i)3-s + (−0.130 − 0.991i)4-s + (0.826 + 0.562i)5-s + (0.132 − 0.991i)6-s − 1.06·7-s + (−0.831 − 0.555i)8-s + (0.385 − 0.922i)9-s + (0.968 − 0.250i)10-s + (−0.452 − 0.328i)11-s + (−0.658 − 0.752i)12-s + (0.558 + 0.768i)13-s + (−0.698 + 0.797i)14-s + (0.999 + 0.0102i)15-s + (−0.965 + 0.258i)16-s + (−0.0837 + 0.257i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0258 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0258 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.63563 - 1.59388i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.63563 - 1.59388i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.932 + 1.06i)T \) |
| 3 | \( 1 + (-1.44 + 0.959i)T \) |
| 5 | \( 1 + (-1.84 - 1.25i)T \) |
| good | 7 | \( 1 + 2.80T + 7T^{2} \) |
| 11 | \( 1 + (1.50 + 1.08i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-2.01 - 2.76i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (0.345 - 1.06i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-7.11 - 2.31i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (1.23 - 1.70i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (3.56 - 1.15i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (8.53 + 2.77i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-4.97 - 6.84i)T + (-11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (3.81 + 5.25i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 7.87T + 43T^{2} \) |
| 47 | \( 1 + (0.999 - 0.324i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.595 - 1.83i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (6.91 - 5.02i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (0.730 + 0.530i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (0.462 - 1.42i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (1.11 + 3.41i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-7.45 + 10.2i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-4.76 + 1.54i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (5.84 + 1.89i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-9.96 + 13.7i)T + (-27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (16.7 - 5.45i)T + (78.4 - 57.0i)T^{2} \) |
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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
−11.65043005202928096246175225489, −10.56853482881728343198443078670, −9.557778936400970731357092875528, −9.178700478525685238092490531552, −7.46531007251823855727829192085, −6.40094001082619612298767707004, −5.65311110037978617180678256398, −3.70769219620836019733382586213, −2.96569989549095387149910931508, −1.67820404361773520057827709785,
2.64792813457148411151131991424, 3.66917005928542203470942040026, 5.03770589468486870709694446478, 5.80395866738064786595582561376, 7.13835949305976455613310297448, 8.100482794064649046753740058532, 9.259853050809671907777465588017, 9.633212638490300176266001721929, 10.94511204944921828686859410615, 12.56505805689059997938648756271
