| L(s) = 1 | + (3.02 + 0.810i)2-s + (0.887 + 2.86i)3-s + (5.02 + 2.90i)4-s + (2.83 − 4.12i)5-s + (0.360 + 9.38i)6-s + (−5.96 − 3.66i)7-s + (3.99 + 3.99i)8-s + (−7.42 + 5.08i)9-s + (11.9 − 10.1i)10-s + (−10.3 − 5.96i)11-s + (−3.85 + 16.9i)12-s + (7.11 + 7.11i)13-s + (−15.0 − 15.9i)14-s + (14.3 + 4.46i)15-s + (−2.75 − 4.77i)16-s + (−3.02 + 0.811i)17-s + ⋯ |
| L(s) = 1 | + (1.51 + 0.405i)2-s + (0.295 + 0.955i)3-s + (1.25 + 0.725i)4-s + (0.566 − 0.824i)5-s + (0.0600 + 1.56i)6-s + (−0.852 − 0.523i)7-s + (0.499 + 0.499i)8-s + (−0.825 + 0.564i)9-s + (1.19 − 1.01i)10-s + (−0.939 − 0.542i)11-s + (−0.321 + 1.41i)12-s + (0.547 + 0.547i)13-s + (−1.07 − 1.13i)14-s + (0.954 + 0.297i)15-s + (−0.172 − 0.298i)16-s + (−0.178 + 0.0477i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.641 - 0.767i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.641 - 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.65364 + 1.24046i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.65364 + 1.24046i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.887 - 2.86i)T \) |
| 5 | \( 1 + (-2.83 + 4.12i)T \) |
| 7 | \( 1 + (5.96 + 3.66i)T \) |
| good | 2 | \( 1 + (-3.02 - 0.810i)T + (3.46 + 2i)T^{2} \) |
| 11 | \( 1 + (10.3 + 5.96i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-7.11 - 7.11i)T + 169iT^{2} \) |
| 17 | \( 1 + (3.02 - 0.811i)T + (250. - 144.5i)T^{2} \) |
| 19 | \( 1 + (-17.2 - 29.8i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-2.51 + 9.38i)T + (-458. - 264.5i)T^{2} \) |
| 29 | \( 1 - 26.5T + 841T^{2} \) |
| 31 | \( 1 + (-0.677 - 0.391i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (7.62 - 28.4i)T + (-1.18e3 - 684.5i)T^{2} \) |
| 41 | \( 1 - 43.0T + 1.68e3T^{2} \) |
| 43 | \( 1 + (40.7 - 40.7i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-7.76 + 28.9i)T + (-1.91e3 - 1.10e3i)T^{2} \) |
| 53 | \( 1 + (-17.1 + 4.59i)T + (2.43e3 - 1.40e3i)T^{2} \) |
| 59 | \( 1 + (62.8 + 36.2i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-21.4 + 12.3i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (56.2 - 15.0i)T + (3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 - 43.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-0.332 + 0.0891i)T + (4.61e3 - 2.66e3i)T^{2} \) |
| 79 | \( 1 + (-66.7 + 38.5i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (76.7 - 76.7i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + (-56.8 + 32.8i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-53.5 + 53.5i)T - 9.40e3iT^{2} \) |
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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
−13.71274818461217790795051416683, −13.07782762937750421826388058550, −11.93682475947177847716537639658, −10.46018752107072465228988136875, −9.487550329200372769719325314717, −8.127939006268424852773180777138, −6.29578774746818144322261416743, −5.37691584226916336810332903228, −4.25294235988083634074333793449, −3.09937942760002302970520494847,
2.45982238198484512196119610369, 3.16772779163305913576503550851, 5.34825650859015531013890324774, 6.29583417186843396719415881075, 7.30935649482805421640785917239, 9.114127165605592440502760019345, 10.58742206183522922892289054914, 11.68327262610156626419648275114, 12.71699216887089679034970872464, 13.36028252934696790055769980658
