L(s) = 1 | + (−0.349 + 1.37i)2-s + (1.72 − 0.203i)3-s + (−1.75 − 0.958i)4-s + 2.27i·5-s + (−0.323 + 2.42i)6-s − i·7-s + (1.92 − 2.06i)8-s + (2.91 − 0.699i)9-s + (−3.12 − 0.796i)10-s − 4.31·11-s + (−3.21 − 1.29i)12-s − 0.406·13-s + (1.37 + 0.349i)14-s + (0.463 + 3.91i)15-s + (2.16 + 3.36i)16-s − 4.31i·17-s + ⋯ |
L(s) = 1 | + (−0.247 + 0.968i)2-s + (0.993 − 0.117i)3-s + (−0.877 − 0.479i)4-s + 1.01i·5-s + (−0.131 + 0.991i)6-s − 0.377i·7-s + (0.681 − 0.731i)8-s + (0.972 − 0.233i)9-s + (−0.986 − 0.251i)10-s − 1.30·11-s + (−0.927 − 0.372i)12-s − 0.112·13-s + (0.366 + 0.0934i)14-s + (0.119 + 1.01i)15-s + (0.540 + 0.841i)16-s − 1.04i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.372 - 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.372 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.849393 + 0.574020i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.849393 + 0.574020i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.349 - 1.37i)T \) |
| 3 | \( 1 + (-1.72 + 0.203i)T \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 - 2.27iT - 5T^{2} \) |
| 11 | \( 1 + 4.31T + 11T^{2} \) |
| 13 | \( 1 + 0.406T + 13T^{2} \) |
| 17 | \( 1 + 4.31iT - 17T^{2} \) |
| 19 | \( 1 + 5.42iT - 19T^{2} \) |
| 23 | \( 1 + 1.16T + 23T^{2} \) |
| 29 | \( 1 - 3.72iT - 29T^{2} \) |
| 31 | \( 1 - 5.83iT - 31T^{2} \) |
| 37 | \( 1 + 7.83T + 37T^{2} \) |
| 41 | \( 1 + 2.56iT - 41T^{2} \) |
| 43 | \( 1 - 11.0iT - 43T^{2} \) |
| 47 | \( 1 + 2.32T + 47T^{2} \) |
| 53 | \( 1 - 5.48iT - 53T^{2} \) |
| 59 | \( 1 + 3.91T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 - 7.83iT - 67T^{2} \) |
| 71 | \( 1 - 4.88T + 71T^{2} \) |
| 73 | \( 1 - 2.81T + 73T^{2} \) |
| 79 | \( 1 + 4iT - 79T^{2} \) |
| 83 | \( 1 + 0.641T + 83T^{2} \) |
| 89 | \( 1 + 13.9iT - 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−14.47601160065199627150576747381, −13.81072111422622862672882914165, −12.88876994675212721382356739896, −10.80304478055909980747446199793, −9.888801881211033837564874993614, −8.673074881721534202761236489754, −7.45276404863617877655865107052, −6.85420002856986833498818634566, −4.90590518229268249723945115346, −3.00019171085947013424334186529,
2.08163777518713281410239250840, 3.80724863967067408017053414971, 5.22592067556480678949996007130, 7.960482561781990206096208100771, 8.488145298019278568451730508472, 9.675441879848941430984320756691, 10.54629571822235753086296957679, 12.22491786958803004923473089555, 12.88661283218103650883650722046, 13.72720244916694097779471348624