L(s) = 1 | + (0.979 + 1.02i)2-s + (−1.71 + 0.218i)3-s + (−0.0819 + 1.99i)4-s + (2.22 − 0.218i)5-s + (−1.90 − 1.53i)6-s + 7-s + (−2.11 + 1.87i)8-s + (2.90 − 0.750i)9-s + (2.40 + 2.05i)10-s + 2.59·11-s + (−0.295 − 3.45i)12-s + 5.21i·13-s + (0.979 + 1.02i)14-s + (−3.77 + 0.861i)15-s + (−3.98 − 0.327i)16-s − 3.91·17-s + ⋯ |
L(s) = 1 | + (0.692 + 0.721i)2-s + (−0.992 + 0.126i)3-s + (−0.0409 + 0.999i)4-s + (0.995 − 0.0976i)5-s + (−0.777 − 0.628i)6-s + 0.377·7-s + (−0.749 + 0.662i)8-s + (0.968 − 0.250i)9-s + (0.759 + 0.650i)10-s + 0.781·11-s + (−0.0853 − 0.996i)12-s + 1.44i·13-s + (0.261 + 0.272i)14-s + (−0.974 + 0.222i)15-s + (−0.996 − 0.0819i)16-s − 0.950·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.182 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.182 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.10945 + 1.33403i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.10945 + 1.33403i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.979 - 1.02i)T \) |
| 3 | \( 1 + (1.71 - 0.218i)T \) |
| 5 | \( 1 + (-2.22 + 0.218i)T \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 - 2.59T + 11T^{2} \) |
| 13 | \( 1 - 5.21iT - 13T^{2} \) |
| 17 | \( 1 + 3.91T + 17T^{2} \) |
| 19 | \( 1 + 5.21iT - 19T^{2} \) |
| 23 | \( 1 - 7.44iT - 23T^{2} \) |
| 29 | \( 1 - 8.44iT - 29T^{2} \) |
| 31 | \( 1 + 1.42iT - 31T^{2} \) |
| 37 | \( 1 + 7.99iT - 37T^{2} \) |
| 41 | \( 1 + 7.49iT - 41T^{2} \) |
| 43 | \( 1 - 3.28T + 43T^{2} \) |
| 47 | \( 1 + 4.36iT - 47T^{2} \) |
| 53 | \( 1 - 0.310T + 53T^{2} \) |
| 59 | \( 1 + 2.17T + 59T^{2} \) |
| 61 | \( 1 - 6.70T + 61T^{2} \) |
| 67 | \( 1 - 14.3T + 67T^{2} \) |
| 71 | \( 1 + 5.88T + 71T^{2} \) |
| 73 | \( 1 + 11.3iT - 73T^{2} \) |
| 79 | \( 1 + 7.09iT - 79T^{2} \) |
| 83 | \( 1 + 0.802iT - 83T^{2} \) |
| 89 | \( 1 - 6.98iT - 89T^{2} \) |
| 97 | \( 1 + 6.80iT - 97T^{2} \) |
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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
−11.49286666431704355254803238146, −10.89980178975499591366324499949, −9.305486278690453066730663966686, −8.997223881351169175711744626826, −7.11024217581255002174713300741, −6.73838318641314203125650216241, −5.67809319422464289640862654311, −4.89412811224710683157896080055, −3.94110034864038945558377342196, −1.92114222347485137952244185020,
1.14023481561304572068002995646, 2.48974785559688378490205471580, 4.17117108062342194642143039871, 5.14454551320265774801144702618, 6.06848285214074272244615458367, 6.59765706815757340646106284616, 8.253230405081215523571043585342, 9.676455328432419301527980732963, 10.27214824858695038497677818400, 10.98440757213585539422311511993