| L(s) = 1 | + (4.28 − 3.11i)3-s + (1.95 + 6.02i)5-s + (19.9 + 14.5i)7-s + (−66.4 + 204. i)9-s + (243. + 319. i)11-s + (281. − 866. i)13-s + (27.1 + 19.7i)15-s + (694. + 2.13e3i)17-s + (−499. + 363. i)19-s + 130.·21-s + 3.15e3·23-s + (2.49e3 − 1.81e3i)25-s + (749. + 2.30e3i)27-s + (1.10 + 0.801i)29-s + (−2.26e3 + 6.97e3i)31-s + ⋯ |
| L(s) = 1 | + (0.274 − 0.199i)3-s + (0.0350 + 0.107i)5-s + (0.154 + 0.111i)7-s + (−0.273 + 0.841i)9-s + (0.606 + 0.794i)11-s + (0.462 − 1.42i)13-s + (0.0311 + 0.0226i)15-s + (0.583 + 1.79i)17-s + (−0.317 + 0.230i)19-s + 0.0646·21-s + 1.24·23-s + (0.798 − 0.580i)25-s + (0.197 + 0.608i)27-s + (0.000243 + 0.000177i)29-s + (−0.423 + 1.30i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.90392 + 0.692497i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.90392 + 0.692497i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + (-243. - 319. i)T \) |
| good | 3 | \( 1 + (-4.28 + 3.11i)T + (75.0 - 231. i)T^{2} \) |
| 5 | \( 1 + (-1.95 - 6.02i)T + (-2.52e3 + 1.83e3i)T^{2} \) |
| 7 | \( 1 + (-19.9 - 14.5i)T + (5.19e3 + 1.59e4i)T^{2} \) |
| 13 | \( 1 + (-281. + 866. i)T + (-3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-694. - 2.13e3i)T + (-1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (499. - 363. i)T + (7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 - 3.15e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + (-1.10 - 0.801i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (2.26e3 - 6.97e3i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-1.26e4 - 9.15e3i)T + (2.14e7 + 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-6.29e3 + 4.57e3i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 + 6.76e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (1.69e4 - 1.23e4i)T + (7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-6.99e3 + 2.15e4i)T + (-3.38e8 - 2.45e8i)T^{2} \) |
| 59 | \( 1 + (3.24e4 + 2.35e4i)T + (2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (4.48e3 + 1.38e4i)T + (-6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 + 5.23e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (3.19e3 + 9.82e3i)T + (-1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 + (2.33e4 + 1.69e4i)T + (6.40e8 + 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-5.83e3 + 1.79e4i)T + (-2.48e9 - 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-2.14e4 - 6.60e4i)T + (-3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 - 4.44e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (1.09e4 - 3.37e4i)T + (-6.94e9 - 5.04e9i)T^{2} \) |
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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
−13.13113167881197123141323907692, −12.52727408289166516518893377132, −10.98153082896280891674958843312, −10.20162162593034159008301107111, −8.619136456700244421463818199729, −7.81700495166982189607368271579, −6.33993217623059665772544960420, −4.93608782164850016701613140386, −3.19358105469012585642250975345, −1.51462134597661254323763199787,
0.921617934018553966664252305508, 3.03962627670994237889049000885, 4.43101993335903968989208993871, 6.08476492512442031298472763413, 7.28872571844119474627767616870, 9.019521134127884355363893724384, 9.315274659547381936401612507527, 11.18098949277265285895161574860, 11.75733694904705485951203147192, 13.24283942781625398559394343011
