L(s) = 1 | + (−2.71 − 4.70i)2-s + (4.53 + 2.53i)3-s + (−10.7 + 18.6i)4-s + 13.1·5-s + (−0.382 − 28.2i)6-s + (12.4 + 13.7i)7-s + 73.2·8-s + (14.1 + 23.0i)9-s + (−35.6 − 61.6i)10-s − 18.2·11-s + (−95.9 + 57.1i)12-s + (−12.9 − 22.3i)13-s + (30.6 − 95.7i)14-s + (59.4 + 33.2i)15-s + (−112. − 195. i)16-s + (1.04 + 1.80i)17-s + ⋯ |
L(s) = 1 | + (−0.959 − 1.66i)2-s + (0.872 + 0.488i)3-s + (−1.34 + 2.32i)4-s + 1.17·5-s + (−0.0260 − 1.91i)6-s + (0.672 + 0.740i)7-s + 3.23·8-s + (0.523 + 0.852i)9-s + (−1.12 − 1.95i)10-s − 0.500·11-s + (−2.30 + 1.37i)12-s + (−0.275 − 0.477i)13-s + (0.585 − 1.82i)14-s + (1.02 + 0.572i)15-s + (−1.76 − 3.05i)16-s + (0.0148 + 0.0257i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.607 + 0.794i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.607 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.19129 - 0.588338i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.19129 - 0.588338i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-4.53 - 2.53i)T \) |
| 7 | \( 1 + (-12.4 - 13.7i)T \) |
good | 2 | \( 1 + (2.71 + 4.70i)T + (-4 + 6.92i)T^{2} \) |
| 5 | \( 1 - 13.1T + 125T^{2} \) |
| 11 | \( 1 + 18.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + (12.9 + 22.3i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (-1.04 - 1.80i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-10.2 + 17.8i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 - 68.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + (-132. + 229. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-15.2 + 26.3i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (143. - 249. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (152. + 263. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (119. - 207. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (209. + 363. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (114. + 198. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (399. - 692. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (47.1 + 81.6i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-284. + 492. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 211.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (314. + 544. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-167. - 289. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-163. + 282. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-160. + 278. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (60.9 - 105. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.83009623763507305323302299602, −13.11563746418378710814363820554, −11.81603477220010285715827447114, −10.49924397381039064631127079159, −9.821360387812451717847869952093, −8.856368806457064834471303428831, −7.956730566090499919183906826630, −4.89777672978159766591547260041, −2.91835219813142273948114212443, −1.90155751974991857296507162862,
1.47622568325114222751674887792, 5.01984871231602295820752153492, 6.53634053673999826077642151984, 7.46471946227157240213601920753, 8.551078364773910440189264643438, 9.536652236280952219692833219407, 10.49414587349283807735593039804, 13.12932294799433074346107587942, 14.16000865596483833084593380926, 14.41820869858095000091649888165