| 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
−17.86908077189143152359041280590, −16.19082485029223829573694505535, −15.79402146610156973837326187728, −14.36723839585763885471236749814, −11.87149770280988745783000934691, −9.940783389242098991158526564392, −9.413867785663053464452541982116, −8.273551928646497375923883638125, −6.20300989449759518230144144892, −2.59614178813065633708191574193,
0.62258501170052250370502911776, 2.49233565489550572212141611636, 7.20166398981831565602780213626, 7.78458296246078206249304138134, 9.641641885334015109207360516686, 10.42477616910001390996489008594, 12.53469191219292999720667914427, 13.73359122480995840557722399350, 15.80589952704596742162658053275, 17.21223135453386329433792981829
