L(s) = 1 | + (0.707 − 0.707i)2-s + (1.47 − 0.901i)3-s − 1.00i·4-s + (−1.61 + 1.54i)5-s + (0.408 − 1.68i)6-s + (−1.95 − 1.95i)7-s + (−0.707 − 0.707i)8-s + (1.37 − 2.66i)9-s + (−0.0553 + 2.23i)10-s − 3.67i·11-s + (−0.901 − 1.47i)12-s + (2.60 − 2.60i)13-s − 2.76·14-s + (−1.00 + 3.74i)15-s − 1.00·16-s + (−1.52 + 1.52i)17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (0.853 − 0.520i)3-s − 0.500i·4-s + (−0.724 + 0.689i)5-s + (0.166 − 0.687i)6-s + (−0.737 − 0.737i)7-s + (−0.250 − 0.250i)8-s + (0.457 − 0.888i)9-s + (−0.0175 + 0.706i)10-s − 1.10i·11-s + (−0.260 − 0.426i)12-s + (0.722 − 0.722i)13-s − 0.737·14-s + (−0.259 + 0.965i)15-s − 0.250·16-s + (−0.370 + 0.370i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.474 + 0.880i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.474 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.00931 - 1.69075i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.00931 - 1.69075i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.47 + 0.901i)T \) |
| 5 | \( 1 + (1.61 - 1.54i)T \) |
| 19 | \( 1 + iT \) |
good | 7 | \( 1 + (1.95 + 1.95i)T + 7iT^{2} \) |
| 11 | \( 1 + 3.67iT - 11T^{2} \) |
| 13 | \( 1 + (-2.60 + 2.60i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.52 - 1.52i)T - 17iT^{2} \) |
| 23 | \( 1 + (-0.483 - 0.483i)T + 23iT^{2} \) |
| 29 | \( 1 - 8.02T + 29T^{2} \) |
| 31 | \( 1 + 3.06T + 31T^{2} \) |
| 37 | \( 1 + (-7.32 - 7.32i)T + 37iT^{2} \) |
| 41 | \( 1 - 5.46iT - 41T^{2} \) |
| 43 | \( 1 + (3.11 - 3.11i)T - 43iT^{2} \) |
| 47 | \( 1 + (3.44 - 3.44i)T - 47iT^{2} \) |
| 53 | \( 1 + (-7.51 - 7.51i)T + 53iT^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 - 0.989T + 61T^{2} \) |
| 67 | \( 1 + (-8.05 - 8.05i)T + 67iT^{2} \) |
| 71 | \( 1 + 4.07iT - 71T^{2} \) |
| 73 | \( 1 + (-11.1 + 11.1i)T - 73iT^{2} \) |
| 79 | \( 1 - 7.15iT - 79T^{2} \) |
| 83 | \( 1 + (2.66 + 2.66i)T + 83iT^{2} \) |
| 89 | \( 1 + 1.05T + 89T^{2} \) |
| 97 | \( 1 + (-5.51 - 5.51i)T + 97iT^{2} \) |
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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
−10.60341955726341384943889794893, −9.732535496496129743880228102506, −8.528199234740233697472151175235, −7.86621791897109000802879640270, −6.69887130430836465784345820321, −6.16996194975893442523683694741, −4.32410320658723415839917238787, −3.35171980206110780779577640378, −2.89936168754194465862803712290, −0.905225086769090714396627070943,
2.25549843357139624042387206885, 3.57825094622571060900795140791, 4.36976718911214446390622091930, 5.24400865132986400961061644521, 6.59159281693953621747172231854, 7.49980498728774906284281320770, 8.494038111400247881604682931126, 9.073268015804632996877552856544, 9.824364753724226860469803309567, 11.12319212143070390125632131607