L(s) = 1 | + (0.707 − 0.707i)2-s + (−0.162 − 1.72i)3-s − 1.00i·4-s + (−0.520 − 2.17i)5-s + (−1.33 − 1.10i)6-s + (0.496 + 0.496i)7-s + (−0.707 − 0.707i)8-s + (−2.94 + 0.561i)9-s + (−1.90 − 1.17i)10-s − 1.60i·11-s + (−1.72 + 0.162i)12-s + (0.465 − 0.465i)13-s + 0.702·14-s + (−3.66 + 1.25i)15-s − 1.00·16-s + (0.134 − 0.134i)17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (−0.0939 − 0.995i)3-s − 0.500i·4-s + (−0.232 − 0.972i)5-s + (−0.544 − 0.450i)6-s + (0.187 + 0.187i)7-s + (−0.250 − 0.250i)8-s + (−0.982 + 0.187i)9-s + (−0.602 − 0.370i)10-s − 0.482i·11-s + (−0.497 + 0.0469i)12-s + (0.128 − 0.128i)13-s + 0.187·14-s + (−0.946 + 0.322i)15-s − 0.250·16-s + (0.0326 − 0.0326i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0735741 - 1.51628i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0735741 - 1.51628i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 + (0.162 + 1.72i)T \) |
| 5 | \( 1 + (0.520 + 2.17i)T \) |
| 19 | \( 1 + iT \) |
good | 7 | \( 1 + (-0.496 - 0.496i)T + 7iT^{2} \) |
| 11 | \( 1 + 1.60iT - 11T^{2} \) |
| 13 | \( 1 + (-0.465 + 0.465i)T - 13iT^{2} \) |
| 17 | \( 1 + (-0.134 + 0.134i)T - 17iT^{2} \) |
| 23 | \( 1 + (0.307 + 0.307i)T + 23iT^{2} \) |
| 29 | \( 1 - 2.93T + 29T^{2} \) |
| 31 | \( 1 + 1.63T + 31T^{2} \) |
| 37 | \( 1 + (-4.03 - 4.03i)T + 37iT^{2} \) |
| 41 | \( 1 + 3.54iT - 41T^{2} \) |
| 43 | \( 1 + (-3.15 + 3.15i)T - 43iT^{2} \) |
| 47 | \( 1 + (4.45 - 4.45i)T - 47iT^{2} \) |
| 53 | \( 1 + (2.36 + 2.36i)T + 53iT^{2} \) |
| 59 | \( 1 - 8.58T + 59T^{2} \) |
| 61 | \( 1 - 6.10T + 61T^{2} \) |
| 67 | \( 1 + (6.28 + 6.28i)T + 67iT^{2} \) |
| 71 | \( 1 + 13.9iT - 71T^{2} \) |
| 73 | \( 1 + (-2.24 + 2.24i)T - 73iT^{2} \) |
| 79 | \( 1 + 3.48iT - 79T^{2} \) |
| 83 | \( 1 + (-1.76 - 1.76i)T + 83iT^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 + (0.0173 + 0.0173i)T + 97iT^{2} \) |
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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
−10.59080725472849516981343309874, −9.308907689895683455577216226487, −8.507180993144085693788620016602, −7.73470653786711417992940042761, −6.51202316275547076118238223167, −5.59849683668080403170000746642, −4.75279419149130025855575700837, −3.39380761132725163186314031766, −2.03884186260625424163844181981, −0.75129128341386053329252230567,
2.62496700401671085889086051381, 3.73326556676062427782797869066, 4.50510148149651035976960171958, 5.63614459624523235354178973248, 6.53492745019296829130127282212, 7.50659707014515904406937398158, 8.413866538350851337703872208733, 9.550588133865868346367556603399, 10.32019047956176978835203177704, 11.17347901735450196411217682868