L(s) = 1 | + (2.88 − 1.66i)2-s + (16.8 + 21.1i)3-s + (−26.4 + 45.8i)4-s + (−166. + 96.0i)5-s + (83.7 + 32.9i)6-s + (157. − 304. i)7-s + 389. i·8-s + (−163. + 710. i)9-s + (−319. + 554. i)10-s + (750. + 433. i)11-s + (−1.41e3 + 212. i)12-s + 3.41e3·13-s + (−54.6 − 1.14e3i)14-s + (−4.82e3 − 1.89e3i)15-s + (−1.04e3 − 1.80e3i)16-s + (2.06e3 + 1.19e3i)17-s + ⋯ |
L(s) = 1 | + (0.360 − 0.208i)2-s + (0.623 + 0.782i)3-s + (−0.413 + 0.715i)4-s + (−1.33 + 0.768i)5-s + (0.387 + 0.152i)6-s + (0.457 − 0.888i)7-s + 0.760i·8-s + (−0.223 + 0.974i)9-s + (−0.319 + 0.554i)10-s + (0.564 + 0.325i)11-s + (−0.817 + 0.122i)12-s + 1.55·13-s + (−0.0199 − 0.415i)14-s + (−1.43 − 0.562i)15-s + (−0.254 − 0.441i)16-s + (0.420 + 0.242i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.133 - 0.991i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.133 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(1.08827 + 1.24423i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.08827 + 1.24423i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-16.8 - 21.1i)T \) |
| 7 | \( 1 + (-157. + 304. i)T \) |
good | 2 | \( 1 + (-2.88 + 1.66i)T + (32 - 55.4i)T^{2} \) |
| 5 | \( 1 + (166. - 96.0i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 11 | \( 1 + (-750. - 433. i)T + (8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 - 3.41e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + (-2.06e3 - 1.19e3i)T + (1.20e7 + 2.09e7i)T^{2} \) |
| 19 | \( 1 + (1.29e3 + 2.24e3i)T + (-2.35e7 + 4.07e7i)T^{2} \) |
| 23 | \( 1 + (6.49e3 - 3.74e3i)T + (7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 - 2.45e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + (-1.02e4 + 1.76e4i)T + (-4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 + (-3.39e4 - 5.88e4i)T + (-1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 + 8.76e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 1.12e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + (-1.05e5 + 6.11e4i)T + (5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 + (-1.45e5 - 8.37e4i)T + (1.10e10 + 1.91e10i)T^{2} \) |
| 59 | \( 1 + (1.39e5 + 8.02e4i)T + (2.10e10 + 3.65e10i)T^{2} \) |
| 61 | \( 1 + (-1.99e4 - 3.45e4i)T + (-2.57e10 + 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-3.69e4 + 6.39e4i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 - 9.29e3iT - 1.28e11T^{2} \) |
| 73 | \( 1 + (7.80e4 - 1.35e5i)T + (-7.56e10 - 1.31e11i)T^{2} \) |
| 79 | \( 1 + (1.16e5 + 2.02e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + 8.72e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (1.82e5 - 1.05e5i)T + (2.48e11 - 4.30e11i)T^{2} \) |
| 97 | \( 1 - 1.18e6T + 8.32e11T^{2} \) |
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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
−16.97139820211715358813112072135, −15.73118073294709933702658375077, −14.55028649812545983539744275920, −13.54167994165677322151107088110, −11.68082002212746334168332590848, −10.67918751994161017593305819734, −8.595417297170575128260439965630, −7.52441668455268023996637409541, −4.22166940332899213443283336686, −3.49889545392934515288921569278,
1.00904530914749889118660781595, 4.01357697773943408470302540114, 6.03058935560503243431184842894, 8.115554223515237471114247934906, 9.004800816804453306266068473584, 11.52619375021977580344792667704, 12.64130956275297825874419339213, 13.96464951224914717453214277335, 15.10043469348825320578132741474, 16.06583726148979067950611803129