| L(s) = 1 | + (4.29 + 3.12i)2-s + (−1.93 − 5.96i)3-s + (6.24 + 19.2i)4-s + (−9.58 − 5.75i)5-s + (10.2 − 31.6i)6-s − 9.63·7-s + (−20.0 + 61.6i)8-s + (−9.96 + 7.24i)9-s + (−23.2 − 54.6i)10-s + (19.6 + 14.3i)11-s + (102. − 74.5i)12-s + (35.7 − 25.9i)13-s + (−41.4 − 30.0i)14-s + (−15.7 + 68.3i)15-s + (−147. + 107. i)16-s + (0.568 − 1.75i)17-s + ⋯ |
| L(s) = 1 | + (1.51 + 1.10i)2-s + (−0.372 − 1.14i)3-s + (0.780 + 2.40i)4-s + (−0.857 − 0.514i)5-s + (0.700 − 2.15i)6-s − 0.520·7-s + (−0.885 + 2.72i)8-s + (−0.369 + 0.268i)9-s + (−0.734 − 1.72i)10-s + (0.539 + 0.392i)11-s + (2.46 − 1.79i)12-s + (0.762 − 0.554i)13-s + (−0.790 − 0.574i)14-s + (−0.271 + 1.17i)15-s + (−2.31 + 1.67i)16-s + (0.00811 − 0.0249i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 - 0.627i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.778 - 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.78313 + 0.628645i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.78313 + 0.628645i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (9.58 + 5.75i)T \) |
| good | 2 | \( 1 + (-4.29 - 3.12i)T + (2.47 + 7.60i)T^{2} \) |
| 3 | \( 1 + (1.93 + 5.96i)T + (-21.8 + 15.8i)T^{2} \) |
| 7 | \( 1 + 9.63T + 343T^{2} \) |
| 11 | \( 1 + (-19.6 - 14.3i)T + (411. + 1.26e3i)T^{2} \) |
| 13 | \( 1 + (-35.7 + 25.9i)T + (678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-0.568 + 1.75i)T + (-3.97e3 - 2.88e3i)T^{2} \) |
| 19 | \( 1 + (22.1 - 68.2i)T + (-5.54e3 - 4.03e3i)T^{2} \) |
| 23 | \( 1 + (56.7 + 41.2i)T + (3.75e3 + 1.15e4i)T^{2} \) |
| 29 | \( 1 + (31.0 + 95.6i)T + (-1.97e4 + 1.43e4i)T^{2} \) |
| 31 | \( 1 + (51.1 - 157. i)T + (-2.41e4 - 1.75e4i)T^{2} \) |
| 37 | \( 1 + (-316. + 229. i)T + (1.56e4 - 4.81e4i)T^{2} \) |
| 41 | \( 1 + (2.89 - 2.10i)T + (2.12e4 - 6.55e4i)T^{2} \) |
| 43 | \( 1 - 9.64T + 7.95e4T^{2} \) |
| 47 | \( 1 + (167. + 515. i)T + (-8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-50.2 - 154. i)T + (-1.20e5 + 8.75e4i)T^{2} \) |
| 59 | \( 1 + (134. - 97.6i)T + (6.34e4 - 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-692. - 502. i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 + (218. - 671. i)T + (-2.43e5 - 1.76e5i)T^{2} \) |
| 71 | \( 1 + (209. + 644. i)T + (-2.89e5 + 2.10e5i)T^{2} \) |
| 73 | \( 1 + (-39.1 - 28.4i)T + (1.20e5 + 3.69e5i)T^{2} \) |
| 79 | \( 1 + (-145. - 448. i)T + (-3.98e5 + 2.89e5i)T^{2} \) |
| 83 | \( 1 + (185. - 572. i)T + (-4.62e5 - 3.36e5i)T^{2} \) |
| 89 | \( 1 + (558. + 406. i)T + (2.17e5 + 6.70e5i)T^{2} \) |
| 97 | \( 1 + (190. + 585. i)T + (-7.38e5 + 5.36e5i)T^{2} \) |
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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
−16.77507525360369345537176054294, −15.90026168181467252044798300008, −14.70937758976228442054237029827, −13.24907530941615669634549381299, −12.57652551734478045448267600045, −11.72775495926396437540802892905, −8.194390632249457776132376931476, −7.05408625876975487585229422866, −5.88804508237748569112383263748, −3.94350988106668507160396590500,
3.44714522958065633275445861733, 4.47707200632211495859973607277, 6.28744435979204827396181593619, 9.660992224339289587414503109111, 11.01951176126481907243175054766, 11.50456241483284736638340075900, 13.06729314952174835661769250171, 14.44153671746692812386187070542, 15.45807441326804196444107902690, 16.28505007400702455548874497769
