| L(s) = 1 | + (0.909 − 1.08i)2-s + (1.41 + 3.89i)3-s + (−0.347 − 1.96i)4-s + (−0.197 + 1.11i)5-s + (5.51 + 2.00i)6-s + (−5.55 − 9.62i)7-s + (−2.44 − 1.41i)8-s + (−6.29 + 5.28i)9-s + (1.03 + 1.22i)10-s + (3.53 − 6.11i)11-s + (7.18 − 4.14i)12-s + (−6.32 + 17.3i)13-s + (−15.4 − 2.72i)14-s + (−4.63 + 0.817i)15-s + (−3.75 + 1.36i)16-s + (0.827 + 0.694i)17-s + ⋯ |
| L(s) = 1 | + (0.454 − 0.541i)2-s + (0.473 + 1.29i)3-s + (−0.0868 − 0.492i)4-s + (−0.0394 + 0.223i)5-s + (0.919 + 0.334i)6-s + (−0.793 − 1.37i)7-s + (−0.306 − 0.176i)8-s + (−0.699 + 0.586i)9-s + (0.103 + 0.122i)10-s + (0.320 − 0.555i)11-s + (0.598 − 0.345i)12-s + (−0.486 + 1.33i)13-s + (−1.10 − 0.194i)14-s + (−0.309 + 0.0544i)15-s + (−0.234 + 0.0855i)16-s + (0.0487 + 0.0408i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0457i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0457i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.34648 + 0.0307989i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.34648 + 0.0307989i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.909 + 1.08i)T \) |
| 19 | \( 1 + (-0.856 - 18.9i)T \) |
| good | 3 | \( 1 + (-1.41 - 3.89i)T + (-6.89 + 5.78i)T^{2} \) |
| 5 | \( 1 + (0.197 - 1.11i)T + (-23.4 - 8.55i)T^{2} \) |
| 7 | \( 1 + (5.55 + 9.62i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-3.53 + 6.11i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (6.32 - 17.3i)T + (-129. - 108. i)T^{2} \) |
| 17 | \( 1 + (-0.827 - 0.694i)T + (50.1 + 284. i)T^{2} \) |
| 23 | \( 1 + (7.48 + 42.4i)T + (-497. + 180. i)T^{2} \) |
| 29 | \( 1 + (-17.7 - 21.1i)T + (-146. + 828. i)T^{2} \) |
| 31 | \( 1 + (30.5 - 17.6i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 31.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (1.56 + 4.28i)T + (-1.28e3 + 1.08e3i)T^{2} \) |
| 43 | \( 1 + (4.60 - 26.1i)T + (-1.73e3 - 632. i)T^{2} \) |
| 47 | \( 1 + (-19.3 + 16.2i)T + (383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 + (21.7 - 3.83i)T + (2.63e3 - 960. i)T^{2} \) |
| 59 | \( 1 + (-26.0 + 31.0i)T + (-604. - 3.42e3i)T^{2} \) |
| 61 | \( 1 + (-1.55 - 8.83i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (58.1 + 69.3i)T + (-779. + 4.42e3i)T^{2} \) |
| 71 | \( 1 + (-124. - 21.9i)T + (4.73e3 + 1.72e3i)T^{2} \) |
| 73 | \( 1 + (23.2 - 8.46i)T + (4.08e3 - 3.42e3i)T^{2} \) |
| 79 | \( 1 + (26.5 + 73.0i)T + (-4.78e3 + 4.01e3i)T^{2} \) |
| 83 | \( 1 + (40.0 + 69.2i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-17.6 + 48.3i)T + (-6.06e3 - 5.09e3i)T^{2} \) |
| 97 | \( 1 + (-29.7 + 35.5i)T + (-1.63e3 - 9.26e3i)T^{2} \) |
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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
−16.29470034310055771080839429960, −14.48228356400528072706732386410, −14.16828068800506510566919060346, −12.55547646635842049354915405703, −10.85467143698096180177936196632, −10.14412682376078967015676343531, −9.007158916106551686946947447085, −6.73534882901487799901423067128, −4.45161676441548238361579413162, −3.45379355299287463769195009407,
2.73583258219777456137129795849, 5.53656908649885655505271470273, 6.91430311251628692064043555004, 8.122256296548748805787691794489, 9.422952713228001908048596663025, 11.96645511513120499147550332142, 12.71429870552556241533984153457, 13.48290402432277900549631806884, 14.98341002890438466062282604985, 15.72067505050948871709862941561
