| L(s) = 1 | + (0.0694 − 0.660i)2-s + (−0.106 − 1.72i)3-s + (1.52 + 0.324i)4-s + (−0.116 − 1.11i)5-s + (−1.14 − 0.0494i)6-s + (−2.61 + 2.90i)7-s + (0.730 − 2.24i)8-s + (−2.97 + 0.369i)9-s − 0.741·10-s + (3.27 − 0.529i)11-s + (0.397 − 2.67i)12-s + (2.44 − 1.08i)13-s + (1.73 + 1.92i)14-s + (−1.90 + 0.320i)15-s + (1.41 + 0.629i)16-s + (−4.48 + 3.26i)17-s + ⋯ |
| L(s) = 1 | + (0.0491 − 0.467i)2-s + (−0.0617 − 0.998i)3-s + (0.762 + 0.162i)4-s + (−0.0521 − 0.496i)5-s + (−0.469 − 0.0201i)6-s + (−0.987 + 1.09i)7-s + (0.258 − 0.794i)8-s + (−0.992 + 0.123i)9-s − 0.234·10-s + (0.987 − 0.159i)11-s + (0.114 − 0.770i)12-s + (0.678 − 0.302i)13-s + (0.463 + 0.515i)14-s + (−0.492 + 0.0827i)15-s + (0.353 + 0.157i)16-s + (−1.08 + 0.790i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.269 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.269 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.880020 - 0.667569i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.880020 - 0.667569i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.106 + 1.72i)T \) |
| 11 | \( 1 + (-3.27 + 0.529i)T \) |
| good | 2 | \( 1 + (-0.0694 + 0.660i)T + (-1.95 - 0.415i)T^{2} \) |
| 5 | \( 1 + (0.116 + 1.11i)T + (-4.89 + 1.03i)T^{2} \) |
| 7 | \( 1 + (2.61 - 2.90i)T + (-0.731 - 6.96i)T^{2} \) |
| 13 | \( 1 + (-2.44 + 1.08i)T + (8.69 - 9.66i)T^{2} \) |
| 17 | \( 1 + (4.48 - 3.26i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (1.58 - 4.88i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (1.05 - 1.83i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.724 + 0.804i)T + (-3.03 - 28.8i)T^{2} \) |
| 31 | \( 1 + (0.392 - 0.174i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (2.69 + 8.27i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (1.95 + 2.16i)T + (-4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (-1.11 - 1.93i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (12.5 - 2.66i)T + (42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (4.56 + 3.31i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-2.32 - 0.493i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (3.73 + 1.66i)T + (40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (-4.87 + 8.44i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (5.11 - 3.71i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (3.60 + 11.1i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (0.471 - 4.48i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (-0.440 - 0.196i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 - 16.0T + 89T^{2} \) |
| 97 | \( 1 + (0.837 - 7.97i)T + (-94.8 - 20.1i)T^{2} \) |
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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
−13.14818690815257584648104021109, −12.55799939232271117670042438346, −11.88110003965426271214989885799, −10.80871219817338658313199688628, −9.192064918988024159836157364758, −8.210559780416000734794979238992, −6.58415662742868378658944845397, −6.01441497869218565115246389586, −3.45234382984802251463294474804, −1.85679005266241377224377463436,
3.13763102774179688164421073977, 4.57182189711335337749094925009, 6.48490928735631937137981527951, 6.87798374421829808369593406355, 8.768466961516443697203612404046, 9.961038075904788624216134255353, 10.91144387074864848237372056851, 11.58152078019380316519268280251, 13.40647198326233655921784565549, 14.35411493450058654244865452283
