| L(s) = 1 | + (−9.29 − 0.356i)2-s + (−40.0 + 16.1i)3-s + (22.4 + 1.72i)4-s + (101. + 171. i)5-s + (377. − 135. i)6-s + (−487. − 154. i)7-s + (383. + 44.2i)8-s + (816. − 785. i)9-s + (−884. − 1.62e3i)10-s + (−692. − 908. i)11-s + (−925. + 292. i)12-s + (−2.18e3 + 423. i)13-s + (4.47e3 + 1.60e3i)14-s + (−6.83e3 − 5.20e3i)15-s + (−4.97e3 − 767. i)16-s + (3.22e3 − 4.98e3i)17-s + ⋯ |
| L(s) = 1 | + (−1.16 − 0.0445i)2-s + (−1.48 + 0.597i)3-s + (0.350 + 0.0268i)4-s + (0.814 + 1.36i)5-s + (1.74 − 0.627i)6-s + (−1.42 − 0.449i)7-s + (0.749 + 0.0864i)8-s + (1.11 − 1.07i)9-s + (−0.884 − 1.62i)10-s + (−0.520 − 0.682i)11-s + (−0.535 + 0.169i)12-s + (−0.994 + 0.192i)13-s + (1.63 + 0.585i)14-s + (−2.02 − 1.54i)15-s + (−1.21 − 0.187i)16-s + (0.656 − 1.01i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.625 - 0.780i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.625 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.266978 + 0.128229i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.266978 + 0.128229i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 83 | \( 1 + (3.94e5 - 4.13e5i)T \) |
| good | 2 | \( 1 + (9.29 + 0.356i)T + (63.8 + 4.89i)T^{2} \) |
| 3 | \( 1 + (40.0 - 16.1i)T + (525. - 505. i)T^{2} \) |
| 5 | \( 1 + (-101. - 171. i)T + (-7.46e3 + 1.37e4i)T^{2} \) |
| 7 | \( 1 + (487. + 154. i)T + (9.62e4 + 6.76e4i)T^{2} \) |
| 11 | \( 1 + (692. + 908. i)T + (-4.69e5 + 1.70e6i)T^{2} \) |
| 13 | \( 1 + (2.18e3 - 423. i)T + (4.47e6 - 1.80e6i)T^{2} \) |
| 17 | \( 1 + (-3.22e3 + 4.98e3i)T + (-9.87e6 - 2.20e7i)T^{2} \) |
| 19 | \( 1 + (-757. + 3.23e3i)T + (-4.21e7 - 2.08e7i)T^{2} \) |
| 23 | \( 1 + (-3.76e3 + 4.22e3i)T + (-1.69e7 - 1.47e8i)T^{2} \) |
| 29 | \( 1 + (-2.33e4 - 1.15e4i)T + (3.60e8 + 4.73e8i)T^{2} \) |
| 31 | \( 1 + (-1.09e3 - 9.50e3i)T + (-8.64e8 + 2.02e8i)T^{2} \) |
| 37 | \( 1 + (6.74e4 + 6.48e4i)T + (9.82e7 + 2.56e9i)T^{2} \) |
| 41 | \( 1 + (-4.77e3 - 1.24e5i)T + (-4.73e9 + 3.63e8i)T^{2} \) |
| 43 | \( 1 + (6.45e4 - 2.89e4i)T + (4.20e9 - 4.71e9i)T^{2} \) |
| 47 | \( 1 + (3.69e4 + 4.48e4i)T + (-2.05e9 + 1.05e10i)T^{2} \) |
| 53 | \( 1 + (-2.04e4 + 2.48e4i)T + (-4.21e9 - 2.17e10i)T^{2} \) |
| 59 | \( 1 + (-3.80e4 - 1.38e5i)T + (-3.62e10 + 2.15e10i)T^{2} \) |
| 61 | \( 1 + (-1.42e5 + 1.00e5i)T + (1.74e10 - 4.84e10i)T^{2} \) |
| 67 | \( 1 + (-8.04e3 + 5.20e4i)T + (-8.62e10 - 2.72e10i)T^{2} \) |
| 71 | \( 1 + (5.22e4 + 1.65e5i)T + (-1.04e11 + 7.36e10i)T^{2} \) |
| 73 | \( 1 + (6.13e5 - 3.33e5i)T + (8.22e10 - 1.27e11i)T^{2} \) |
| 79 | \( 1 + (-5.40e4 + 7.04e5i)T + (-2.40e11 - 3.71e10i)T^{2} \) |
| 89 | \( 1 + (-7.55e5 + 2.71e5i)T + (3.83e11 - 3.16e11i)T^{2} \) |
| 97 | \( 1 + (-1.11e6 - 3.99e5i)T + (6.42e11 + 5.29e11i)T^{2} \) |
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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
−13.19751415452018020873207282414, −11.63403347867923493427000046653, −10.46888303378621303507507484106, −10.21728837557357816394810564977, −9.389101340408195005359803666656, −7.17031026174243743327317341895, −6.46781877933552424101731691969, −5.09866960414787940820538108392, −2.97183467856817277510918428970, −0.49143655982547917521784437411,
0.46214855279511511711826747713, 1.71035680490936628718818932942, 4.94960796187367377471461581470, 5.88688471125202037278878261679, 7.13505993688730872072202763434, 8.568009284865360179040543542811, 9.890371802886501500147108403442, 10.16583942069019453632149548961, 12.15300870341797922378968097958, 12.62534480640874187485404182946
