| L(s) = 1 | + (1.41 − 1.41i)2-s + (0.0236 − 3.99i)4-s + (7.58 − 2.75i)5-s + (9.72 − 1.71i)7-s + (−5.60 − 5.70i)8-s + (6.86 − 14.6i)10-s + (−5.60 + 15.4i)11-s + (4.71 + 3.95i)13-s + (11.3 − 16.1i)14-s + (−15.9 − 0.189i)16-s + (−4.02 + 6.96i)17-s + (2.87 − 1.66i)19-s + (−10.8 − 30.3i)20-s + (13.7 + 29.7i)22-s + (−19.7 − 3.47i)23-s + ⋯ |
| L(s) = 1 | + (0.709 − 0.705i)2-s + (0.00592 − 0.999i)4-s + (1.51 − 0.551i)5-s + (1.38 − 0.244i)7-s + (−0.700 − 0.713i)8-s + (0.686 − 1.46i)10-s + (−0.509 + 1.40i)11-s + (0.362 + 0.304i)13-s + (0.812 − 1.15i)14-s + (−0.999 − 0.0118i)16-s + (−0.236 + 0.409i)17-s + (0.151 − 0.0874i)19-s + (−0.542 − 1.51i)20-s + (0.625 + 1.35i)22-s + (−0.857 − 0.151i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.47238 - 2.09146i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.47238 - 2.09146i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.41 + 1.41i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-7.58 + 2.75i)T + (19.1 - 16.0i)T^{2} \) |
| 7 | \( 1 + (-9.72 + 1.71i)T + (46.0 - 16.7i)T^{2} \) |
| 11 | \( 1 + (5.60 - 15.4i)T + (-92.6 - 77.7i)T^{2} \) |
| 13 | \( 1 + (-4.71 - 3.95i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (4.02 - 6.96i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-2.87 + 1.66i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (19.7 + 3.47i)T + (497. + 180. i)T^{2} \) |
| 29 | \( 1 + (13.6 - 11.4i)T + (146. - 828. i)T^{2} \) |
| 31 | \( 1 + (26.6 + 4.69i)T + (903. + 328. i)T^{2} \) |
| 37 | \( 1 + (-7.47 + 12.9i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (17.2 + 14.4i)T + (291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-17.9 + 49.3i)T + (-1.41e3 - 1.18e3i)T^{2} \) |
| 47 | \( 1 + (61.7 - 10.8i)T + (2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 + 18.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-12.4 - 34.1i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-15.9 - 90.6i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (3.38 - 4.03i)T + (-779. - 4.42e3i)T^{2} \) |
| 71 | \( 1 + (-47.7 - 27.5i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-26.0 - 45.1i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (38.8 + 46.3i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (79.5 + 94.7i)T + (-1.19e3 + 6.78e3i)T^{2} \) |
| 89 | \( 1 + (-87.9 - 152. i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (36.4 + 13.2i)T + (7.20e3 + 6.04e3i)T^{2} \) |
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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
−11.16710364988705654094283145057, −10.33167224797964623379820563899, −9.622240611996777566730400196028, −8.621080513561423199543934216152, −7.16668423026999882085590088515, −5.82289856562915154741507512634, −5.07651972822087397231391443426, −4.22741912647729378195219427055, −2.13405009056786869249929285878, −1.60216651213907430960537944756,
2.00904761595741886286200117110, 3.25751837954351671037623970104, 4.96325899554842329779965564237, 5.70015731210034102144046784970, 6.40154537696852180889248930045, 7.80291826596957498962930354625, 8.499796786951962040946563109905, 9.656481854561132198516764443059, 10.98823358867578821168213153181, 11.43077558893232835064274825995
