| L(s) = 1 | + (10.4 + 12.1i)2-s + (−89.5 − 29.1i)3-s + (−37.3 + 253. i)4-s + (−624. + 19.7i)5-s + (−584. − 1.38e3i)6-s + 1.00e3i·7-s + (−3.45e3 + 2.19e3i)8-s + (1.87e3 + 1.35e3i)9-s + (−6.77e3 − 7.35e3i)10-s + (−5.70e3 − 7.85e3i)11-s + (1.07e4 − 2.16e4i)12-s + (−3.17e4 − 2.30e4i)13-s + (−1.21e4 + 1.04e4i)14-s + (5.65e4 + 1.64e4i)15-s + (−6.27e4 − 1.88e4i)16-s + (1.74e4 + 5.36e4i)17-s + ⋯ |
| L(s) = 1 | + (0.653 + 0.756i)2-s + (−1.10 − 0.359i)3-s + (−0.145 + 0.989i)4-s + (−0.999 + 0.0315i)5-s + (−0.450 − 1.07i)6-s + 0.418i·7-s + (−0.844 + 0.536i)8-s + (0.285 + 0.207i)9-s + (−0.677 − 0.735i)10-s + (−0.389 − 0.536i)11-s + (0.516 − 1.04i)12-s + (−1.11 − 0.806i)13-s + (−0.316 + 0.273i)14-s + (1.11 + 0.324i)15-s + (−0.957 − 0.288i)16-s + (0.208 + 0.641i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 - 0.349i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.937 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.651499 + 0.117426i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.651499 + 0.117426i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-10.4 - 12.1i)T \) |
| 5 | \( 1 + (624. - 19.7i)T \) |
| good | 3 | \( 1 + (89.5 + 29.1i)T + (5.30e3 + 3.85e3i)T^{2} \) |
| 7 | \( 1 - 1.00e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + (5.70e3 + 7.85e3i)T + (-6.62e7 + 2.03e8i)T^{2} \) |
| 13 | \( 1 + (3.17e4 + 2.30e4i)T + (2.52e8 + 7.75e8i)T^{2} \) |
| 17 | \( 1 + (-1.74e4 - 5.36e4i)T + (-5.64e9 + 4.10e9i)T^{2} \) |
| 19 | \( 1 + (1.68e5 - 5.46e4i)T + (1.37e10 - 9.98e9i)T^{2} \) |
| 23 | \( 1 + (4.83e4 + 6.65e4i)T + (-2.41e10 + 7.44e10i)T^{2} \) |
| 29 | \( 1 + (1.57e5 - 4.83e5i)T + (-4.04e11 - 2.94e11i)T^{2} \) |
| 31 | \( 1 + (-7.53e5 + 2.44e5i)T + (6.90e11 - 5.01e11i)T^{2} \) |
| 37 | \( 1 + (-3.00e6 - 2.18e6i)T + (1.08e12 + 3.34e12i)T^{2} \) |
| 41 | \( 1 + (-2.51e6 - 1.83e6i)T + (2.46e12 + 7.59e12i)T^{2} \) |
| 43 | \( 1 + 5.08e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + (4.55e6 + 1.48e6i)T + (1.92e13 + 1.39e13i)T^{2} \) |
| 53 | \( 1 + (-1.79e6 + 5.51e6i)T + (-5.03e13 - 3.65e13i)T^{2} \) |
| 59 | \( 1 + (1.32e6 - 1.82e6i)T + (-4.53e13 - 1.39e14i)T^{2} \) |
| 61 | \( 1 + (1.41e7 - 1.03e7i)T + (5.92e13 - 1.82e14i)T^{2} \) |
| 67 | \( 1 + (2.79e7 - 9.08e6i)T + (3.28e14 - 2.38e14i)T^{2} \) |
| 71 | \( 1 + (7.40e6 + 2.40e6i)T + (5.22e14 + 3.79e14i)T^{2} \) |
| 73 | \( 1 + (-7.52e6 + 5.46e6i)T + (2.49e14 - 7.66e14i)T^{2} \) |
| 79 | \( 1 + (-2.50e7 - 8.15e6i)T + (1.22e15 + 8.91e14i)T^{2} \) |
| 83 | \( 1 + (3.74e7 - 1.21e7i)T + (1.82e15 - 1.32e15i)T^{2} \) |
| 89 | \( 1 + (-3.63e7 + 2.64e7i)T + (1.21e15 - 3.74e15i)T^{2} \) |
| 97 | \( 1 + (1.64e7 - 5.05e7i)T + (-6.34e15 - 4.60e15i)T^{2} \) |
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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
−12.35691214109987170508842244851, −11.68245918702448838090086615564, −10.55692694664899147380674278918, −8.538853549477346390217686983399, −7.68299611972497728650641452723, −6.46390276319628335038818762276, −5.55719103015045702624466663570, −4.43383347965888471492303992186, −2.90842595908839662809473369042, −0.33889508883835016515907345335,
0.56209168565604209518581253481, 2.52045142995448748789801539548, 4.33631442079618562895509967024, 4.73472164556306865643858116991, 6.23517253364404004665459173831, 7.52695548731256672593658131229, 9.401477862862442751150810529772, 10.51218774157925514177765350763, 11.30355717195902813047549089841, 12.01545882521602717993827611643
