L(s) = 1 | + (2.06e6 + 3.55e5i)2-s + 9.12e9i·3-s + (4.14e12 + 1.46e12i)4-s − 1.85e14·5-s + (−3.24e15 + 1.88e16i)6-s + 7.39e17i·7-s + (8.04e18 + 4.51e18i)8-s + 2.61e19·9-s + (−3.84e20 − 6.60e19i)10-s − 7.81e20i·11-s + (−1.34e22 + 3.78e22i)12-s − 1.76e23·13-s + (−2.63e23 + 1.52e24i)14-s − 1.69e24i·15-s + (1.50e25 + 1.21e25i)16-s + 1.77e25·17-s + ⋯ |
L(s) = 1 | + (0.985 + 0.169i)2-s + 0.872i·3-s + (0.942 + 0.334i)4-s − 0.389·5-s + (−0.147 + 0.859i)6-s + 1.32i·7-s + (0.872 + 0.489i)8-s + 0.239·9-s + (−0.384 − 0.0660i)10-s − 0.105i·11-s + (−0.291 + 0.822i)12-s − 0.713·13-s + (−0.224 + 1.30i)14-s − 0.339i·15-s + (0.776 + 0.629i)16-s + 0.257·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.942 - 0.334i)\, \overline{\Lambda}(43-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+21) \, L(s)\cr =\mathstrut & (-0.942 - 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{43}{2})\) |
\(\approx\) |
\(3.219568890\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.219568890\) |
\(L(22)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-2.06e6 - 3.55e5i)T \) |
good | 3 | \( 1 - 9.12e9iT - 1.09e20T^{2} \) |
| 5 | \( 1 + 1.85e14T + 2.27e29T^{2} \) |
| 7 | \( 1 - 7.39e17iT - 3.11e35T^{2} \) |
| 11 | \( 1 + 7.81e20iT - 5.47e43T^{2} \) |
| 13 | \( 1 + 1.76e23T + 6.10e46T^{2} \) |
| 17 | \( 1 - 1.77e25T + 4.77e51T^{2} \) |
| 19 | \( 1 - 8.99e26iT - 5.10e53T^{2} \) |
| 23 | \( 1 + 2.55e28iT - 1.55e57T^{2} \) |
| 29 | \( 1 + 1.56e30T + 2.63e61T^{2} \) |
| 31 | \( 1 + 3.90e31iT - 4.33e62T^{2} \) |
| 37 | \( 1 - 2.18e32T + 7.31e65T^{2} \) |
| 41 | \( 1 + 1.09e34T + 5.45e67T^{2} \) |
| 43 | \( 1 - 3.18e34iT - 4.03e68T^{2} \) |
| 47 | \( 1 - 2.91e34iT - 1.69e70T^{2} \) |
| 53 | \( 1 + 2.53e36T + 2.62e72T^{2} \) |
| 59 | \( 1 + 2.40e36iT - 2.37e74T^{2} \) |
| 61 | \( 1 - 4.12e37T + 9.63e74T^{2} \) |
| 67 | \( 1 - 3.27e37iT - 4.95e76T^{2} \) |
| 71 | \( 1 + 5.97e38iT - 5.66e77T^{2} \) |
| 73 | \( 1 - 8.93e38T + 1.81e78T^{2} \) |
| 79 | \( 1 - 7.24e39iT - 5.01e79T^{2} \) |
| 83 | \( 1 - 1.04e40iT - 3.99e80T^{2} \) |
| 89 | \( 1 - 1.28e41T + 7.48e81T^{2} \) |
| 97 | \( 1 - 5.53e41T + 2.78e83T^{2} \) |
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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
−15.47833868193069560074917151437, −14.66803450972796596306815673895, −12.66489057313141726330227714582, −11.53858162821743795966241675485, −9.814829697491281436922374423871, −7.921950394157079108967582818760, −6.00266586811925285748098261257, −4.77793476529342131588221462290, −3.54506070486837218452383444613, −2.10728881857780187029686475283,
0.59777837763148835093837636945, 1.82775407987288592398449177584, 3.52111905070989101773680258475, 4.84336972645396004996850674960, 6.83400859728766604029641575251, 7.49944175359751651210244423731, 10.27380684398293932141630718573, 11.77547875493695766279022229153, 13.05216056860468220372227861090, 14.01790158034436133037047780444