L(s) = 1 | + (0.599 + 1.28i)2-s + (0.468 − 1.66i)3-s + (−1.28 + 1.53i)4-s + i·5-s + (2.41 − 0.400i)6-s − 0.936i·7-s + (−2.73 − 0.719i)8-s + (−2.56 − 1.56i)9-s + (−1.28 + 0.599i)10-s − 4.27·11-s + (1.96 + 2.85i)12-s + 3.12·13-s + (1.19 − 0.561i)14-s + (1.66 + 0.468i)15-s + (−0.719 − 3.93i)16-s − 2i·17-s + ⋯ |
L(s) = 1 | + (0.424 + 0.905i)2-s + (0.270 − 0.962i)3-s + (−0.640 + 0.768i)4-s + 0.447i·5-s + (0.986 − 0.163i)6-s − 0.353i·7-s + (−0.967 − 0.254i)8-s + (−0.853 − 0.520i)9-s + (−0.405 + 0.189i)10-s − 1.28·11-s + (0.566 + 0.824i)12-s + 0.866·13-s + (0.320 − 0.150i)14-s + (0.430 + 0.120i)15-s + (−0.179 − 0.983i)16-s − 0.485i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 - 0.566i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.824 - 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.978736 + 0.303865i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.978736 + 0.303865i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.599 - 1.28i)T \) |
| 3 | \( 1 + (-0.468 + 1.66i)T \) |
| 5 | \( 1 - iT \) |
good | 7 | \( 1 + 0.936iT - 7T^{2} \) |
| 11 | \( 1 + 4.27T + 11T^{2} \) |
| 13 | \( 1 - 3.12T + 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 - 4.27iT - 19T^{2} \) |
| 23 | \( 1 - 7.60T + 23T^{2} \) |
| 29 | \( 1 - 5.12iT - 29T^{2} \) |
| 31 | \( 1 - 2.39iT - 31T^{2} \) |
| 37 | \( 1 + 3.12T + 37T^{2} \) |
| 41 | \( 1 + 7.12iT - 41T^{2} \) |
| 43 | \( 1 + 1.46iT - 43T^{2} \) |
| 47 | \( 1 + 0.936T + 47T^{2} \) |
| 53 | \( 1 + 4.24iT - 53T^{2} \) |
| 59 | \( 1 + 7.19T + 59T^{2} \) |
| 61 | \( 1 + 5.12T + 61T^{2} \) |
| 67 | \( 1 + 5.20iT - 67T^{2} \) |
| 71 | \( 1 - 6.67T + 71T^{2} \) |
| 73 | \( 1 + 8.24T + 73T^{2} \) |
| 79 | \( 1 + 9.06iT - 79T^{2} \) |
| 83 | \( 1 + 4.68T + 83T^{2} \) |
| 89 | \( 1 - 6.24iT - 89T^{2} \) |
| 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
−15.07545303584686628180404336943, −14.00710363071771586558179372761, −13.28666480287548796363482657370, −12.31425150702233224132889434805, −10.76511867283110920050760973735, −8.859104192445492659709432432835, −7.72522086622339893024831848000, −6.81823875920884828348016733970, −5.44369194765993277791341361341, −3.22694146055192446093530899592,
2.87913374273600121785135790634, 4.53852969408277104701046962527, 5.65432451686143326045612141125, 8.394129189816109509280775230956, 9.360767634609036809384467997366, 10.57857601017191980427561196748, 11.38050475721826584363927395409, 12.91423213666856429135643076153, 13.64744692494593608731451465108, 15.12721853998387560758954067671