| L(s) = 1 | + (−0.341 + 0.871i)2-s + (−0.116 + 1.55i)3-s + (0.823 + 0.764i)4-s + (−0.0606 − 0.0413i)5-s + (−1.31 − 0.634i)6-s + (2.53 + 0.767i)7-s + (−2.63 + 1.26i)8-s + (0.550 + 0.0829i)9-s + (0.0567 − 0.0386i)10-s + (−0.988 + 0.149i)11-s + (−1.28 + 1.19i)12-s + (1.07 + 1.35i)13-s + (−1.53 + 1.94i)14-s + (0.0715 − 0.0896i)15-s + (−0.0365 − 0.487i)16-s + (1.37 + 0.425i)17-s + ⋯ |
| L(s) = 1 | + (−0.241 + 0.616i)2-s + (−0.0674 + 0.899i)3-s + (0.411 + 0.382i)4-s + (−0.0271 − 0.0184i)5-s + (−0.538 − 0.259i)6-s + (0.957 + 0.289i)7-s + (−0.931 + 0.448i)8-s + (0.183 + 0.0276i)9-s + (0.0179 − 0.0122i)10-s + (−0.298 + 0.0449i)11-s + (−0.371 + 0.344i)12-s + (0.299 + 0.375i)13-s + (−0.410 + 0.519i)14-s + (0.0184 − 0.0231i)15-s + (−0.00912 − 0.121i)16-s + (0.334 + 0.103i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 - 0.572i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.819 - 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.451095 + 1.43335i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.451095 + 1.43335i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-2.53 - 0.767i)T \) |
| 11 | \( 1 + (0.988 - 0.149i)T \) |
| good | 2 | \( 1 + (0.341 - 0.871i)T + (-1.46 - 1.36i)T^{2} \) |
| 3 | \( 1 + (0.116 - 1.55i)T + (-2.96 - 0.447i)T^{2} \) |
| 5 | \( 1 + (0.0606 + 0.0413i)T + (1.82 + 4.65i)T^{2} \) |
| 13 | \( 1 + (-1.07 - 1.35i)T + (-2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (-1.37 - 0.425i)T + (14.0 + 9.57i)T^{2} \) |
| 19 | \( 1 + (-0.810 - 1.40i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.36 - 0.421i)T + (19.0 - 12.9i)T^{2} \) |
| 29 | \( 1 + (-0.127 - 0.557i)T + (-26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (-3.64 + 6.30i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.55 + 2.36i)T + (2.76 - 36.8i)T^{2} \) |
| 41 | \( 1 + (3.10 - 1.49i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (8.55 + 4.11i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (1.14 - 2.92i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (5.86 + 5.44i)T + (3.96 + 52.8i)T^{2} \) |
| 59 | \( 1 + (-0.0110 + 0.00752i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (-6.00 + 5.57i)T + (4.55 - 60.8i)T^{2} \) |
| 67 | \( 1 + (0.692 - 1.19i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.94 + 12.8i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-1.36 - 3.47i)T + (-53.5 + 49.6i)T^{2} \) |
| 79 | \( 1 + (-6.13 - 10.6i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (9.87 - 12.3i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-9.75 - 1.46i)T + (85.0 + 26.2i)T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
| 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
−11.20053349623154829803372763065, −10.21905363009037615016538993481, −9.373750799280362009021189567979, −8.286620588489705187166414895210, −7.83444513932029174175423384880, −6.66215752371227380335427866078, −5.63583824814838754184169424263, −4.65071409210066337078580467147, −3.58351190584715303749291803482, −2.07965523584112023967470703734,
1.02737285488874042781611595021, 1.92590004498841032465211334953, 3.26885721043259158550769476297, 4.84996012749545042782706184324, 5.96312615681817841323149395036, 6.94069695140448376712307741812, 7.70359408839025904779054820788, 8.627689063407535442473764780148, 9.885480162154936700355890703312, 10.51061429305896905263727621376
