L(s) = 1 | + (−1.41 − 0.0184i)2-s + (0.827 + 1.52i)3-s + (1.99 + 0.0520i)4-s + (−1.23 − 1.23i)5-s + (−1.14 − 2.16i)6-s − 7-s + (−2.82 − 0.110i)8-s + (−1.62 + 2.51i)9-s + (1.72 + 1.77i)10-s + (−3.57 + 3.57i)11-s + (1.57 + 3.08i)12-s + (1.58 + 1.58i)13-s + (1.41 + 0.0184i)14-s + (0.859 − 2.91i)15-s + (3.99 + 0.208i)16-s + 2.20i·17-s + ⋯ |
L(s) = 1 | + (−0.999 − 0.0130i)2-s + (0.477 + 0.878i)3-s + (0.999 + 0.0260i)4-s + (−0.554 − 0.554i)5-s + (−0.466 − 0.884i)6-s − 0.377·7-s + (−0.999 − 0.0390i)8-s + (−0.543 + 0.839i)9-s + (0.546 + 0.561i)10-s + (−1.07 + 1.07i)11-s + (0.454 + 0.890i)12-s + (0.440 + 0.440i)13-s + (0.377 + 0.00491i)14-s + (0.221 − 0.751i)15-s + (0.998 + 0.0520i)16-s + 0.535i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.743 - 0.668i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.743 - 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.199749 + 0.521263i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.199749 + 0.521263i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.41 + 0.0184i)T \) |
| 3 | \( 1 + (-0.827 - 1.52i)T \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + (1.23 + 1.23i)T + 5iT^{2} \) |
| 11 | \( 1 + (3.57 - 3.57i)T - 11iT^{2} \) |
| 13 | \( 1 + (-1.58 - 1.58i)T + 13iT^{2} \) |
| 17 | \( 1 - 2.20iT - 17T^{2} \) |
| 19 | \( 1 + (3.76 - 3.76i)T - 19iT^{2} \) |
| 23 | \( 1 - 3.97iT - 23T^{2} \) |
| 29 | \( 1 + (-4.75 + 4.75i)T - 29iT^{2} \) |
| 31 | \( 1 - 1.24iT - 31T^{2} \) |
| 37 | \( 1 + (6.39 - 6.39i)T - 37iT^{2} \) |
| 41 | \( 1 + 3.93T + 41T^{2} \) |
| 43 | \( 1 + (-1.19 - 1.19i)T + 43iT^{2} \) |
| 47 | \( 1 - 8.26T + 47T^{2} \) |
| 53 | \( 1 + (-2.18 - 2.18i)T + 53iT^{2} \) |
| 59 | \( 1 + (-5.45 + 5.45i)T - 59iT^{2} \) |
| 61 | \( 1 + (5.10 + 5.10i)T + 61iT^{2} \) |
| 67 | \( 1 + (7.23 - 7.23i)T - 67iT^{2} \) |
| 71 | \( 1 + 6.13iT - 71T^{2} \) |
| 73 | \( 1 + 10.2iT - 73T^{2} \) |
| 79 | \( 1 + 6.53iT - 79T^{2} \) |
| 83 | \( 1 + (-12.3 - 12.3i)T + 83iT^{2} \) |
| 89 | \( 1 - 9.15T + 89T^{2} \) |
| 97 | \( 1 - 7.67T + 97T^{2} \) |
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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
−11.76863593774668618399704323872, −10.45745051019134167745909318002, −10.17338610307018732868907268255, −9.074618872938941325475086114990, −8.296139705003666551542586867680, −7.62499445899238304299544344396, −6.21362094181015576444338963544, −4.80244473706177446792991292584, −3.60828141603857974411643020363, −2.13714637331642639235373122279,
0.47874272120028432582605844814, 2.54165153734070034801687830189, 3.32663585798186139703551901887, 5.71477840865086728437326878025, 6.77616426867782452674463432119, 7.44735619110163350439581961461, 8.456804100479406175904028985103, 8.930649495154284487387049458783, 10.46604876886850592953309411675, 10.96563522256216969694771750468