| L(s) = 1 | + (−13.8 − 2.44i)2-s + (19.5 − 23.3i)3-s + (125. + 45.7i)4-s + (43.8 − 15.9i)5-s + (−328. + 275. i)6-s + (−286. − 495. i)7-s + (−848. − 490. i)8-s + (−34.5 − 195. i)9-s + (−646. + 114. i)10-s + (−671. + 1.16e3i)11-s + (3.52e3 − 2.03e3i)12-s + (−2.20e3 − 2.62e3i)13-s + (2.75e3 + 7.56e3i)14-s + (486. − 1.33e3i)15-s + (4.00e3 + 3.35e3i)16-s + (108. − 615. i)17-s + ⋯ |
| L(s) = 1 | + (−1.73 − 0.305i)2-s + (0.725 − 0.864i)3-s + (1.96 + 0.714i)4-s + (0.351 − 0.127i)5-s + (−1.51 + 1.27i)6-s + (−0.834 − 1.44i)7-s + (−1.65 − 0.957i)8-s + (−0.0473 − 0.268i)9-s + (−0.646 + 0.114i)10-s + (−0.504 + 0.874i)11-s + (2.04 − 1.17i)12-s + (−1.00 − 1.19i)13-s + (1.00 + 2.75i)14-s + (0.144 − 0.396i)15-s + (0.977 + 0.819i)16-s + (0.0220 − 0.125i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.446i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.894 + 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.142285 - 0.603330i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.142285 - 0.603330i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (-5.41e3 + 4.21e3i)T \) |
| good | 2 | \( 1 + (13.8 + 2.44i)T + (60.1 + 21.8i)T^{2} \) |
| 3 | \( 1 + (-19.5 + 23.3i)T + (-126. - 717. i)T^{2} \) |
| 5 | \( 1 + (-43.8 + 15.9i)T + (1.19e4 - 1.00e4i)T^{2} \) |
| 7 | \( 1 + (286. + 495. i)T + (-5.88e4 + 1.01e5i)T^{2} \) |
| 11 | \( 1 + (671. - 1.16e3i)T + (-8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 + (2.20e3 + 2.62e3i)T + (-8.38e5 + 4.75e6i)T^{2} \) |
| 17 | \( 1 + (-108. + 615. i)T + (-2.26e7 - 8.25e6i)T^{2} \) |
| 23 | \( 1 + (1.10e4 + 4.03e3i)T + (1.13e8 + 9.51e7i)T^{2} \) |
| 29 | \( 1 + (-9.13e3 + 1.61e3i)T + (5.58e8 - 2.03e8i)T^{2} \) |
| 31 | \( 1 + (-2.96e4 + 1.70e4i)T + (4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 + 1.92e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + (-6.17e4 + 7.35e4i)T + (-8.24e8 - 4.67e9i)T^{2} \) |
| 43 | \( 1 + (-8.28e4 + 3.01e4i)T + (4.84e9 - 4.06e9i)T^{2} \) |
| 47 | \( 1 + (8.68e3 + 4.92e4i)T + (-1.01e10 + 3.68e9i)T^{2} \) |
| 53 | \( 1 + (4.05e4 - 1.11e5i)T + (-1.69e10 - 1.42e10i)T^{2} \) |
| 59 | \( 1 + (-2.03e5 - 3.59e4i)T + (3.96e10 + 1.44e10i)T^{2} \) |
| 61 | \( 1 + (8.72e4 + 3.17e4i)T + (3.94e10 + 3.31e10i)T^{2} \) |
| 67 | \( 1 + (-3.25e5 + 5.73e4i)T + (8.50e10 - 3.09e10i)T^{2} \) |
| 71 | \( 1 + (2.00e4 + 5.50e4i)T + (-9.81e10 + 8.23e10i)T^{2} \) |
| 73 | \( 1 + (1.43e5 + 1.20e5i)T + (2.62e10 + 1.49e11i)T^{2} \) |
| 79 | \( 1 + (4.09e4 - 4.87e4i)T + (-4.22e10 - 2.39e11i)T^{2} \) |
| 83 | \( 1 + (-3.27e5 - 5.67e5i)T + (-1.63e11 + 2.83e11i)T^{2} \) |
| 89 | \( 1 + (4.48e5 + 5.34e5i)T + (-8.62e10 + 4.89e11i)T^{2} \) |
| 97 | \( 1 + (-2.35e5 - 4.15e4i)T + (7.82e11 + 2.84e11i)T^{2} \) |
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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
−17.21223135453386329433792981829, −15.80589952704596742162658053275, −13.73359122480995840557722399350, −12.53469191219292999720667914427, −10.42477616910001390996489008594, −9.641641885334015109207360516686, −7.78458296246078206249304138134, −7.20166398981831565602780213626, −2.49233565489550572212141611636, −0.62258501170052250370502911776,
2.59614178813065633708191574193, 6.20300989449759518230144144892, 8.273551928646497375923883638125, 9.413867785663053464452541982116, 9.940783389242098991158526564392, 11.87149770280988745783000934691, 14.36723839585763885471236749814, 15.79402146610156973837326187728, 16.19082485029223829573694505535, 17.86908077189143152359041280590
