| L(s) = 1 | + (3.17 − 6.58i)2-s + (−5.77 + 8.46i)3-s + (−23.3 − 29.3i)4-s + (−12.3 − 13.3i)5-s + (37.4 + 64.8i)6-s + (−52.8 − 30.5i)7-s + (−153. + 34.9i)8-s + (−8.76 − 22.3i)9-s + (−127. + 39.2i)10-s + (110. − 138. i)11-s + (383. − 28.7i)12-s + (162. + 50.1i)13-s + (−368. + 251. i)14-s + (184. − 27.8i)15-s + (−122. + 535. i)16-s + (−266. − 247. i)17-s + ⋯ |
| L(s) = 1 | + (0.793 − 1.64i)2-s + (−0.641 + 0.940i)3-s + (−1.46 − 1.83i)4-s + (−0.495 − 0.534i)5-s + (1.04 + 1.80i)6-s + (−1.07 − 0.622i)7-s + (−2.39 + 0.546i)8-s + (−0.108 − 0.275i)9-s + (−1.27 + 0.392i)10-s + (0.910 − 1.14i)11-s + (2.65 − 0.199i)12-s + (0.961 + 0.296i)13-s + (−1.88 + 1.28i)14-s + (0.820 − 0.123i)15-s + (−0.477 + 2.09i)16-s + (−0.923 − 0.857i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.196i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.980 - 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.108312 + 1.09283i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.108312 + 1.09283i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (1.73e3 + 647. i)T \) |
| good | 2 | \( 1 + (-3.17 + 6.58i)T + (-9.97 - 12.5i)T^{2} \) |
| 3 | \( 1 + (5.77 - 8.46i)T + (-29.5 - 75.4i)T^{2} \) |
| 5 | \( 1 + (12.3 + 13.3i)T + (-46.7 + 623. i)T^{2} \) |
| 7 | \( 1 + (52.8 + 30.5i)T + (1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-110. + 138. i)T + (-3.25e3 - 1.42e4i)T^{2} \) |
| 13 | \( 1 + (-162. - 50.1i)T + (2.35e4 + 1.60e4i)T^{2} \) |
| 17 | \( 1 + (266. + 247. i)T + (6.24e3 + 8.32e4i)T^{2} \) |
| 19 | \( 1 + (-401. - 157. i)T + (9.55e4 + 8.86e4i)T^{2} \) |
| 23 | \( 1 + (469. + 70.7i)T + (2.67e5 + 8.24e4i)T^{2} \) |
| 29 | \( 1 + (342. + 502. i)T + (-2.58e5 + 6.58e5i)T^{2} \) |
| 31 | \( 1 + (-77.9 - 1.04e3i)T + (-9.13e5 + 1.37e5i)T^{2} \) |
| 37 | \( 1 + (-1.33e3 + 768. i)T + (9.37e5 - 1.62e6i)T^{2} \) |
| 41 | \( 1 + (-1.78e3 - 860. i)T + (1.76e6 + 2.20e6i)T^{2} \) |
| 47 | \( 1 + (-794. - 996. i)T + (-1.08e6 + 4.75e6i)T^{2} \) |
| 53 | \( 1 + (-4.31e3 + 1.33e3i)T + (6.51e6 - 4.44e6i)T^{2} \) |
| 59 | \( 1 + (-252. + 1.10e3i)T + (-1.09e7 - 5.25e6i)T^{2} \) |
| 61 | \( 1 + (2.85e3 + 213. i)T + (1.36e7 + 2.06e6i)T^{2} \) |
| 67 | \( 1 + (1.87e3 - 4.77e3i)T + (-1.47e7 - 1.37e7i)T^{2} \) |
| 71 | \( 1 + (-128. - 854. i)T + (-2.42e7 + 7.49e6i)T^{2} \) |
| 73 | \( 1 + (-2.19e3 + 7.13e3i)T + (-2.34e7 - 1.59e7i)T^{2} \) |
| 79 | \( 1 + (1.97e3 - 3.41e3i)T + (-1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 + (790. + 539. i)T + (1.73e7 + 4.41e7i)T^{2} \) |
| 89 | \( 1 + (-6.98e3 + 1.02e4i)T + (-2.29e7 - 5.84e7i)T^{2} \) |
| 97 | \( 1 + (-1.15e3 + 1.45e3i)T + (-1.96e7 - 8.63e7i)T^{2} \) |
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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
−14.02789185288062552933797279833, −13.28228331568965632071307985075, −11.85729515400687406316179782810, −11.21018718466741426872838939890, −10.11380101202874442875941470757, −9.097011443095824128006461429496, −5.96381167191080874763011649735, −4.36588131549750037641603253632, −3.54633957823289282617792076563, −0.62046122258385256241639901228,
3.84290841663468619966161753440, 5.91727156999995676905601257996, 6.62835141225485115599411299606, 7.53977913603586314841349599387, 9.210155237939198365051391175043, 11.70106715378221728775119668184, 12.69958267309531598260008374233, 13.45537946699538311267612089089, 15.02154046662100131304981916994, 15.54993917369238758781500377898
