L(s) = 1 | + (−0.707 + 0.707i)2-s + (−1.41 − 1.41i)3-s − 1.00i·4-s + (−0.707 − 2.12i)5-s + 2.00·6-s + (−2.82 + 2.82i)7-s + (0.707 + 0.707i)8-s + 1.00i·9-s + (2 + 0.999i)10-s − 4·11-s + (−1.41 + 1.41i)12-s + (2 − 2i)13-s − 4.00i·14-s + (−1.99 + 4i)15-s − 1.00·16-s + (5 + 5i)17-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (−0.816 − 0.816i)3-s − 0.500i·4-s + (−0.316 − 0.948i)5-s + 0.816·6-s + (−1.06 + 1.06i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s + (0.632 + 0.316i)10-s − 1.20·11-s + (−0.408 + 0.408i)12-s + (0.554 − 0.554i)13-s − 1.06i·14-s + (−0.516 + 1.03i)15-s − 0.250·16-s + (1.21 + 1.21i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.170 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.170 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.183766 + 0.218372i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.183766 + 0.218372i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (0.707 + 2.12i)T \) |
| 43 | \( 1 + (-0.535 - 6.53i)T \) |
good | 3 | \( 1 + (1.41 + 1.41i)T + 3iT^{2} \) |
| 7 | \( 1 + (2.82 - 2.82i)T - 7iT^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + (-2 + 2i)T - 13iT^{2} \) |
| 17 | \( 1 + (-5 - 5i)T + 17iT^{2} \) |
| 19 | \( 1 - 4.24T + 19T^{2} \) |
| 23 | \( 1 + (5 - 5i)T - 23iT^{2} \) |
| 29 | \( 1 - 4.24T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + (4.24 - 4.24i)T - 37iT^{2} \) |
| 41 | \( 1 + 12T + 41T^{2} \) |
| 47 | \( 1 + (5 + 5i)T + 47iT^{2} \) |
| 53 | \( 1 - 53iT^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 - 4.24iT - 61T^{2} \) |
| 67 | \( 1 + (2 + 2i)T + 67iT^{2} \) |
| 71 | \( 1 - 5.65iT - 71T^{2} \) |
| 73 | \( 1 + (-1.41 - 1.41i)T + 73iT^{2} \) |
| 79 | \( 1 - 10iT - 79T^{2} \) |
| 83 | \( 1 + (-6 + 6i)T - 83iT^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 + (-9 - 9i)T + 97iT^{2} \) |
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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
−11.80492585904344841567055214105, −10.26212128952693825256722729791, −9.643831504772539117619646643162, −8.321981672343759566765583039392, −7.966949031932369193881891996047, −6.61535021826330573440236830483, −5.72050041536769906331092709983, −5.33432562851094761789494607335, −3.28756673497654838920593747001, −1.30918334681467185394194591335,
0.25656437065591258210781784916, 2.87449900030571154165815576195, 3.75415106349849940817914898747, 4.95197989284377843706270277186, 6.29776237429667155659789947250, 7.22267585827641568428322435734, 8.094629354658803420988745614473, 9.733448745853539517941585221933, 10.21848854346127369239996281188, 10.57028597129474452107205629824