L(s) = 1 | + (−1.17 + 0.780i)2-s + (1.51 − 0.848i)3-s + (0.780 − 1.84i)4-s − i·5-s + (−1.11 + 2.17i)6-s + 3.02i·7-s + (0.516 + 2.78i)8-s + (1.56 − 2.56i)9-s + (0.780 + 1.17i)10-s − 1.32·11-s + (−0.382 − 3.44i)12-s − 5.12·13-s + (−2.35 − 3.56i)14-s + (−0.848 − 1.51i)15-s + (−2.78 − 2.87i)16-s + 2i·17-s + ⋯ |
L(s) = 1 | + (−0.833 + 0.552i)2-s + (0.871 − 0.489i)3-s + (0.390 − 0.920i)4-s − 0.447i·5-s + (−0.456 + 0.889i)6-s + 1.14i·7-s + (0.182 + 0.983i)8-s + (0.520 − 0.853i)9-s + (0.246 + 0.372i)10-s − 0.399·11-s + (−0.110 − 0.993i)12-s − 1.42·13-s + (−0.630 − 0.951i)14-s + (−0.218 − 0.389i)15-s + (−0.695 − 0.718i)16-s + 0.485i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.110i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.758574 + 0.0419982i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.758574 + 0.0419982i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.17 - 0.780i)T \) |
| 3 | \( 1 + (-1.51 + 0.848i)T \) |
| 5 | \( 1 + iT \) |
good | 7 | \( 1 - 3.02iT - 7T^{2} \) |
| 11 | \( 1 + 1.32T + 11T^{2} \) |
| 13 | \( 1 + 5.12T + 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 1.32iT - 19T^{2} \) |
| 23 | \( 1 + 0.371T + 23T^{2} \) |
| 29 | \( 1 - 3.12iT - 29T^{2} \) |
| 31 | \( 1 - 4.71iT - 31T^{2} \) |
| 37 | \( 1 - 5.12T + 37T^{2} \) |
| 41 | \( 1 + 1.12iT - 41T^{2} \) |
| 43 | \( 1 + 7.73iT - 43T^{2} \) |
| 47 | \( 1 + 3.02T + 47T^{2} \) |
| 53 | \( 1 + 12.2iT - 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 - 3.12T + 61T^{2} \) |
| 67 | \( 1 - 4.34iT - 67T^{2} \) |
| 71 | \( 1 + 3.39T + 71T^{2} \) |
| 73 | \( 1 - 8.24T + 73T^{2} \) |
| 79 | \( 1 + 8.10iT - 79T^{2} \) |
| 83 | \( 1 + 15.1T + 83T^{2} \) |
| 89 | \( 1 - 10.2iT - 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−15.12407479916955656824875434893, −14.42469899769261099193997714919, −12.90973305905867438056492609908, −11.87238853578059910031331998057, −10.02214145196781651648096060403, −9.005623193342057176763778508592, −8.166632937750527930853992492516, −6.92714077433312159736841806645, −5.30164997060275174402870554670, −2.31676855308356340958263694260,
2.67296548747861602112455215304, 4.25219051623020014989640998550, 7.21832871575891314428271089902, 7.961453812093259537304129142922, 9.611144784286800274692146893400, 10.17765183474206511607747784791, 11.32031436263468872707154234936, 12.89136371949222549899684441440, 13.98433746968497289492782654017, 15.11336137720187425577559621820