| 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
−14.31734659030881169803807458852, −13.75475491756414388063161145697, −12.17782634622418154884019368754, −10.30943957130403827201847474408, −9.071056305556907202440963522015, −8.463406033442953977814498917016, −6.30933293028446591685872215736, −5.31331064577425786123521520450, −3.32972323270482259694014736080, −1.18312823098375168455620443190,
1.47159110320398018048567315447, 2.37557789238742339234112875319, 4.32839486423630788804772608769, 6.36309514012559076784639777407, 7.901251069017484932960518334648, 9.374866030416164103746865226677, 10.39573413346439843578377770107, 11.66587418346043543061849931396, 13.39695053218206272255454178197, 13.69311374863958966646685150711
