| L(s) = 1 | + (1.21 − 8.91i)3-s + (1.79 + 2.14i)5-s + (−63.3 + 23.0i)7-s + (−78.0 − 21.6i)9-s + (−23.1 + 27.5i)11-s + (8.29 + 47.0i)13-s + (21.2 − 13.4i)15-s + (0.309 − 0.178i)17-s + (−163. + 283. i)19-s + (128. + 592. i)21-s + (−242. + 666. i)23-s + (107. − 607. i)25-s + (−287. + 670. i)27-s + (−1.51e3 − 266. i)29-s + (547. + 199. i)31-s + ⋯ |
| L(s) = 1 | + (0.134 − 0.990i)3-s + (0.0718 + 0.0856i)5-s + (−1.29 + 0.470i)7-s + (−0.963 − 0.266i)9-s + (−0.191 + 0.228i)11-s + (0.0490 + 0.278i)13-s + (0.0945 − 0.0596i)15-s + (0.00107 − 0.000619i)17-s + (−0.453 + 0.786i)19-s + (0.292 + 1.34i)21-s + (−0.458 + 1.25i)23-s + (0.171 − 0.972i)25-s + (−0.394 + 0.919i)27-s + (−1.79 − 0.316i)29-s + (0.569 + 0.207i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.687 - 0.725i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.687 - 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.0590743 + 0.137365i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0590743 + 0.137365i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.21 + 8.91i)T \) |
| good | 5 | \( 1 + (-1.79 - 2.14i)T + (-108. + 615. i)T^{2} \) |
| 7 | \( 1 + (63.3 - 23.0i)T + (1.83e3 - 1.54e3i)T^{2} \) |
| 11 | \( 1 + (23.1 - 27.5i)T + (-2.54e3 - 1.44e4i)T^{2} \) |
| 13 | \( 1 + (-8.29 - 47.0i)T + (-2.68e4 + 9.76e3i)T^{2} \) |
| 17 | \( 1 + (-0.309 + 0.178i)T + (4.17e4 - 7.23e4i)T^{2} \) |
| 19 | \( 1 + (163. - 283. i)T + (-6.51e4 - 1.12e5i)T^{2} \) |
| 23 | \( 1 + (242. - 666. i)T + (-2.14e5 - 1.79e5i)T^{2} \) |
| 29 | \( 1 + (1.51e3 + 266. i)T + (6.64e5 + 2.41e5i)T^{2} \) |
| 31 | \( 1 + (-547. - 199. i)T + (7.07e5 + 5.93e5i)T^{2} \) |
| 37 | \( 1 + (343. + 595. i)T + (-9.37e5 + 1.62e6i)T^{2} \) |
| 41 | \( 1 + (498. - 87.9i)T + (2.65e6 - 9.66e5i)T^{2} \) |
| 43 | \( 1 + (1.25e3 + 1.05e3i)T + (5.93e5 + 3.36e6i)T^{2} \) |
| 47 | \( 1 + (-453. - 1.24e3i)T + (-3.73e6 + 3.13e6i)T^{2} \) |
| 53 | \( 1 + 1.43e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + (989. + 1.17e3i)T + (-2.10e6 + 1.19e7i)T^{2} \) |
| 61 | \( 1 + (-4.82e3 + 1.75e3i)T + (1.06e7 - 8.89e6i)T^{2} \) |
| 67 | \( 1 + (474. + 2.69e3i)T + (-1.89e7 + 6.89e6i)T^{2} \) |
| 71 | \( 1 + (4.49e3 - 2.59e3i)T + (1.27e7 - 2.20e7i)T^{2} \) |
| 73 | \( 1 + (-4.36e3 + 7.56e3i)T + (-1.41e7 - 2.45e7i)T^{2} \) |
| 79 | \( 1 + (-814. + 4.61e3i)T + (-3.66e7 - 1.33e7i)T^{2} \) |
| 83 | \( 1 + (9.40e3 + 1.65e3i)T + (4.45e7 + 1.62e7i)T^{2} \) |
| 89 | \( 1 + (9.67e3 + 5.58e3i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 + (-1.33e3 - 1.12e3i)T + (1.53e7 + 8.71e7i)T^{2} \) |
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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
−13.20110678455533785116209719369, −12.50125272314511518754570094799, −11.57484341832782835543859049929, −10.05252365963305351373772576168, −9.026384320594490509566187021109, −7.77787012807504870517209006210, −6.60790037493341606330482596362, −5.72795603945359187637254370377, −3.48994960042902265778255270419, −2.03781820099036212686879010143,
0.06077206114189307942880580515, 2.91054857140261338853160664184, 4.08915715693382273814497490533, 5.55960046936227452390836553786, 6.86375281061793147846745609404, 8.475983566196634951795509170539, 9.518759260153891705471758425047, 10.33056770102825559720961647788, 11.29277657002823481292662179670, 12.80172700658521158815855281422
