| L(s) = 1 | + (−2.77 − 15.7i)2-s + (74.8 − 62.7i)3-s + (−240. + 87.5i)4-s + (1.60e3 + 582. i)5-s + (−1.19e3 − 1.00e3i)6-s + (2.05e3 − 3.56e3i)7-s + (2.04e3 + 3.54e3i)8-s + (−1.76e3 + 9.99e3i)9-s + (4.73e3 − 2.68e4i)10-s + (2.91e4 + 5.05e4i)11-s + (−1.24e4 + 2.16e4i)12-s + (8.55e4 + 7.17e4i)13-s + (−6.18e4 − 2.25e4i)14-s + (1.56e5 − 5.69e4i)15-s + (5.02e4 − 4.21e4i)16-s + (5.14e4 + 2.91e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 − 0.696i)2-s + (0.533 − 0.447i)3-s + (−0.469 + 0.171i)4-s + (1.14 + 0.417i)5-s + (−0.377 − 0.316i)6-s + (0.323 − 0.560i)7-s + (0.176 + 0.306i)8-s + (−0.0895 + 0.507i)9-s + (0.149 − 0.849i)10-s + (0.600 + 1.04i)11-s + (−0.173 + 0.301i)12-s + (0.830 + 0.696i)13-s + (−0.430 − 0.156i)14-s + (0.797 − 0.290i)15-s + (0.191 − 0.160i)16-s + (0.149 + 0.847i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.804 + 0.594i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.804 + 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.55646 - 0.842149i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.55646 - 0.842149i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.77 + 15.7i)T \) |
| 19 | \( 1 + (-7.28e4 + 5.63e5i)T \) |
| good | 3 | \( 1 + (-74.8 + 62.7i)T + (3.41e3 - 1.93e4i)T^{2} \) |
| 5 | \( 1 + (-1.60e3 - 582. i)T + (1.49e6 + 1.25e6i)T^{2} \) |
| 7 | \( 1 + (-2.05e3 + 3.56e3i)T + (-2.01e7 - 3.49e7i)T^{2} \) |
| 11 | \( 1 + (-2.91e4 - 5.05e4i)T + (-1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 + (-8.55e4 - 7.17e4i)T + (1.84e9 + 1.04e10i)T^{2} \) |
| 17 | \( 1 + (-5.14e4 - 2.91e5i)T + (-1.11e11 + 4.05e10i)T^{2} \) |
| 23 | \( 1 + (5.89e5 - 2.14e5i)T + (1.37e12 - 1.15e12i)T^{2} \) |
| 29 | \( 1 + (-9.82e5 + 5.57e6i)T + (-1.36e13 - 4.96e12i)T^{2} \) |
| 31 | \( 1 + (-4.95e6 + 8.57e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 - 4.69e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + (2.20e7 - 1.84e7i)T + (5.68e13 - 3.22e14i)T^{2} \) |
| 43 | \( 1 + (-9.24e5 - 3.36e5i)T + (3.85e14 + 3.23e14i)T^{2} \) |
| 47 | \( 1 + (-1.54e6 + 8.75e6i)T + (-1.05e15 - 3.82e14i)T^{2} \) |
| 53 | \( 1 + (-1.25e7 + 4.55e6i)T + (2.52e15 - 2.12e15i)T^{2} \) |
| 59 | \( 1 + (5.52e6 + 3.13e7i)T + (-8.14e15 + 2.96e15i)T^{2} \) |
| 61 | \( 1 + (1.39e8 - 5.08e7i)T + (8.95e15 - 7.51e15i)T^{2} \) |
| 67 | \( 1 + (2.68e7 - 1.52e8i)T + (-2.55e16 - 9.30e15i)T^{2} \) |
| 71 | \( 1 + (2.49e8 + 9.06e7i)T + (3.51e16 + 2.94e16i)T^{2} \) |
| 73 | \( 1 + (-2.21e8 + 1.85e8i)T + (1.02e16 - 5.79e16i)T^{2} \) |
| 79 | \( 1 + (-2.57e8 + 2.16e8i)T + (2.08e16 - 1.18e17i)T^{2} \) |
| 83 | \( 1 + (6.96e7 - 1.20e8i)T + (-9.34e16 - 1.61e17i)T^{2} \) |
| 89 | \( 1 + (-5.06e8 - 4.24e8i)T + (6.08e16 + 3.45e17i)T^{2} \) |
| 97 | \( 1 + (3.28e7 + 1.86e8i)T + (-7.14e17 + 2.60e17i)T^{2} \) |
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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
−13.69311374863958966646685150711, −13.39695053218206272255454178197, −11.66587418346043543061849931396, −10.39573413346439843578377770107, −9.374866030416164103746865226677, −7.901251069017484932960518334648, −6.36309514012559076784639777407, −4.32839486423630788804772608769, −2.37557789238742339234112875319, −1.47159110320398018048567315447,
1.18312823098375168455620443190, 3.32972323270482259694014736080, 5.31331064577425786123521520450, 6.30933293028446591685872215736, 8.463406033442953977814498917016, 9.071056305556907202440963522015, 10.30943957130403827201847474408, 12.17782634622418154884019368754, 13.75475491756414388063161145697, 14.31734659030881169803807458852
