L(s) = 1 | + (−0.876 − 1.28i)2-s + (1.67 − 0.444i)3-s + (−0.153 + 0.390i)4-s + (−3.15 − 0.973i)5-s + (−2.03 − 1.76i)6-s + (−2.38 − 1.14i)7-s + (−2.39 + 0.547i)8-s + (2.60 − 1.48i)9-s + (1.51 + 4.91i)10-s + (5.25 − 0.393i)11-s + (−0.0830 + 0.722i)12-s + (1.21 − 2.52i)13-s + (0.616 + 4.06i)14-s + (−5.71 − 0.227i)15-s + (3.41 + 3.17i)16-s + (−0.510 − 0.0769i)17-s + ⋯ |
L(s) = 1 | + (−0.619 − 0.908i)2-s + (0.966 − 0.256i)3-s + (−0.0766 + 0.195i)4-s + (−1.41 − 0.435i)5-s + (−0.832 − 0.719i)6-s + (−0.901 − 0.433i)7-s + (−0.847 + 0.193i)8-s + (0.868 − 0.495i)9-s + (0.478 + 1.55i)10-s + (1.58 − 0.118i)11-s + (−0.0239 + 0.208i)12-s + (0.337 − 0.700i)13-s + (0.164 + 1.08i)14-s + (−1.47 − 0.0586i)15-s + (0.854 + 0.792i)16-s + (−0.123 − 0.0186i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.765 + 0.643i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.765 + 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.291173 - 0.798786i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.291173 - 0.798786i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.67 + 0.444i)T \) |
| 7 | \( 1 + (2.38 + 1.14i)T \) |
good | 2 | \( 1 + (0.876 + 1.28i)T + (-0.730 + 1.86i)T^{2} \) |
| 5 | \( 1 + (3.15 + 0.973i)T + (4.13 + 2.81i)T^{2} \) |
| 11 | \( 1 + (-5.25 + 0.393i)T + (10.8 - 1.63i)T^{2} \) |
| 13 | \( 1 + (-1.21 + 2.52i)T + (-8.10 - 10.1i)T^{2} \) |
| 17 | \( 1 + (0.510 + 0.0769i)T + (16.2 + 5.01i)T^{2} \) |
| 19 | \( 1 + (-1.11 - 0.645i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.326 - 2.16i)T + (-21.9 + 6.77i)T^{2} \) |
| 29 | \( 1 + (-2.30 - 1.84i)T + (6.45 + 28.2i)T^{2} \) |
| 31 | \( 1 + (-3.38 + 1.95i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.07 + 2.74i)T + (-27.1 + 25.1i)T^{2} \) |
| 41 | \( 1 + (1.91 + 8.37i)T + (-36.9 + 17.7i)T^{2} \) |
| 43 | \( 1 + (2.50 - 10.9i)T + (-38.7 - 18.6i)T^{2} \) |
| 47 | \( 1 + (2.89 - 1.97i)T + (17.1 - 43.7i)T^{2} \) |
| 53 | \( 1 + (-10.1 - 3.98i)T + (38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (-2.07 + 0.640i)T + (48.7 - 33.2i)T^{2} \) |
| 61 | \( 1 + (7.96 - 3.12i)T + (44.7 - 41.4i)T^{2} \) |
| 67 | \( 1 + (5.74 + 9.95i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.243 - 0.194i)T + (15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (4.32 - 6.34i)T + (-26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + (-4.99 + 8.64i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.35 + 3.54i)T + (51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (0.296 - 3.95i)T + (-88.0 - 13.2i)T^{2} \) |
| 97 | \( 1 - 6.56iT - 97T^{2} \) |
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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
−12.33570730268885470848346784466, −11.80400309739891255661404763355, −10.56992979832701046385505786373, −9.436113701682442107770984298357, −8.769156076415974824596876706722, −7.70189483876216851003392650727, −6.46207943245297742094032760883, −3.98403376251731732649966249963, −3.20252956021503004622154656755, −1.02689760470799535149308376037,
3.16714904362294898045978348054, 4.09339468355494059358269590761, 6.57483699013946329259149682131, 7.09507776838481189979844386163, 8.365765575640215413639773298230, 8.927156670217642695659709642236, 9.929629910276524268565221004340, 11.62507566580855048309492465240, 12.26887787252351264548144193983, 13.75815107116197614091445884928