| L(s) = 1 | + (−1.63 − 2.13i)2-s + (1.11 + 1.32i)3-s + (−1.35 + 5.03i)4-s + (−2.13 − 0.140i)5-s + (0.986 − 4.54i)6-s + (0.0884 + 1.34i)7-s + (7.98 − 3.30i)8-s + (−0.493 + 2.95i)9-s + (3.19 + 4.78i)10-s + (4.13 + 3.62i)11-s + (−8.17 + 3.85i)12-s + (2.78 + 0.747i)13-s + (2.73 − 2.39i)14-s + (−2.20 − 2.98i)15-s + (−11.0 − 6.39i)16-s + (−4.09 + 0.444i)17-s + ⋯ |
| L(s) = 1 | + (−1.15 − 1.50i)2-s + (0.646 + 0.763i)3-s + (−0.675 + 2.51i)4-s + (−0.955 − 0.0626i)5-s + (0.402 − 1.85i)6-s + (0.0334 + 0.510i)7-s + (2.82 − 1.16i)8-s + (−0.164 + 0.986i)9-s + (1.01 + 1.51i)10-s + (1.24 + 1.09i)11-s + (−2.35 + 1.11i)12-s + (0.773 + 0.207i)13-s + (0.730 − 0.640i)14-s + (−0.569 − 0.769i)15-s + (−2.76 − 1.59i)16-s + (−0.994 + 0.107i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.168i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.168i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.640340 + 0.0544215i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.640340 + 0.0544215i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.11 - 1.32i)T \) |
| 17 | \( 1 + (4.09 - 0.444i)T \) |
| good | 2 | \( 1 + (1.63 + 2.13i)T + (-0.517 + 1.93i)T^{2} \) |
| 5 | \( 1 + (2.13 + 0.140i)T + (4.95 + 0.652i)T^{2} \) |
| 7 | \( 1 + (-0.0884 - 1.34i)T + (-6.94 + 0.913i)T^{2} \) |
| 11 | \( 1 + (-4.13 - 3.62i)T + (1.43 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-2.78 - 0.747i)T + (11.2 + 6.5i)T^{2} \) |
| 19 | \( 1 + (-1.52 - 0.630i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (0.0345 + 0.101i)T + (-18.2 + 14.0i)T^{2} \) |
| 29 | \( 1 + (0.787 - 0.388i)T + (17.6 - 23.0i)T^{2} \) |
| 31 | \( 1 + (-0.607 - 0.692i)T + (-4.04 + 30.7i)T^{2} \) |
| 37 | \( 1 + (1.25 + 6.31i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (3.74 - 7.58i)T + (-24.9 - 32.5i)T^{2} \) |
| 43 | \( 1 + (-0.342 + 2.59i)T + (-41.5 - 11.1i)T^{2} \) |
| 47 | \( 1 + (-12.1 + 3.25i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-2.54 + 6.15i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (7.11 + 5.45i)T + (15.2 + 56.9i)T^{2} \) |
| 61 | \( 1 + (-6.73 + 0.441i)T + (60.4 - 7.96i)T^{2} \) |
| 67 | \( 1 + (1.16 - 0.674i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (2.51 - 0.501i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (-5.63 - 3.76i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (-4.94 + 5.64i)T + (-10.3 - 78.3i)T^{2} \) |
| 83 | \( 1 + (0.745 - 0.572i)T + (21.4 - 80.1i)T^{2} \) |
| 89 | \( 1 + (-0.774 + 0.774i)T - 89iT^{2} \) |
| 97 | \( 1 + (1.47 + 2.99i)T + (-59.0 + 76.9i)T^{2} \) |
| 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
−12.50011052549407716955364028621, −11.68764478898917196824868950807, −10.99464875171935123546162742470, −9.868488441014854293172496739007, −9.029246366677569660350921100686, −8.453543804089927932273988038807, −7.27273177916995972461135239807, −4.33229095554909736725178123087, −3.62749916348768658192710213228, −2.03874360546593235120726957684,
0.937282184250800445409753392972, 3.93964304566529091735972631938, 6.02268972353651096452575562271, 6.88625367046609144344622203900, 7.72057206604104164108832035712, 8.622520620496768598351331008773, 9.164301846261877695288608877679, 10.73570278376153508069283575308, 11.76343395167813455986607457959, 13.62811686813620234875393037898
