L(s) = 1 | + (0.707 − 0.707i)2-s + (1.57 + 0.717i)3-s − 1.00i·4-s + (1.60 − 1.55i)5-s + (1.62 − 0.607i)6-s + (2.55 − 0.673i)7-s + (−0.707 − 0.707i)8-s + (1.97 + 2.26i)9-s + (0.0316 − 2.23i)10-s + (−4.13 − 2.38i)11-s + (0.717 − 1.57i)12-s + (0.348 + 1.30i)13-s + (1.33 − 2.28i)14-s + (3.64 − 1.30i)15-s − 1.00·16-s + (−4.94 − 1.32i)17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (0.910 + 0.414i)3-s − 0.500i·4-s + (0.717 − 0.697i)5-s + (0.662 − 0.247i)6-s + (0.967 − 0.254i)7-s + (−0.250 − 0.250i)8-s + (0.656 + 0.754i)9-s + (0.00999 − 0.707i)10-s + (−1.24 − 0.719i)11-s + (0.207 − 0.455i)12-s + (0.0967 + 0.361i)13-s + (0.356 − 0.610i)14-s + (0.941 − 0.337i)15-s − 0.250·16-s + (−1.19 − 0.321i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.647 + 0.762i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.647 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.65999 - 1.23047i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.65999 - 1.23047i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.57 - 0.717i)T \) |
| 5 | \( 1 + (-1.60 + 1.55i)T \) |
| 7 | \( 1 + (-2.55 + 0.673i)T \) |
good | 11 | \( 1 + (4.13 + 2.38i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.348 - 1.30i)T + (-11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 + (4.94 + 1.32i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-1.62 - 0.939i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.33 - 4.99i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (-4.90 - 8.48i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 2.30T + 31T^{2} \) |
| 37 | \( 1 + (3.39 - 0.910i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (4.61 + 2.66i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.375 - 0.100i)T + (37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (-1.56 + 1.56i)T - 47iT^{2} \) |
| 53 | \( 1 + (0.0260 - 0.0971i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + 1.49T + 59T^{2} \) |
| 61 | \( 1 - 13.4T + 61T^{2} \) |
| 67 | \( 1 + (-2.59 - 2.59i)T + 67iT^{2} \) |
| 71 | \( 1 + 14.1iT - 71T^{2} \) |
| 73 | \( 1 + (-5.55 - 1.48i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 - 3.85iT - 79T^{2} \) |
| 83 | \( 1 + (1.17 - 4.36i)T + (-71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (7.07 - 12.2i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.94 - 10.9i)T + (-84.0 - 48.5i)T^{2} \) |
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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
−10.55732519275097859982543239947, −9.646270936234394846318765742611, −8.752979775905528961165517243462, −8.178987606867533326030598752943, −6.96839666917687323453075059697, −5.35263473741826383351267749379, −4.97749615258468282391483141902, −3.82554940789560133690584501003, −2.56860832656798750042763843075, −1.55759711659664769476965847613,
2.11810738104152055881061037220, 2.70251167968567720241927182405, 4.23280145099971946106738449636, 5.23622799217213361275424539263, 6.35665224831804858794222079412, 7.18069297001576650962879147546, 8.023723957018278209438240093380, 8.654926601699968936813039674983, 9.840810488293587032097971576627, 10.60918753652406698020326167807