L(s) = 1 | + (1.61 − 0.433i)2-s + (−0.552 − 0.318i)3-s + (0.700 − 0.404i)4-s + (1.42 − 1.42i)5-s + (−1.03 − 0.276i)6-s + (−1.11 + 2.39i)7-s + (−1.41 + 1.41i)8-s + (−1.29 − 2.24i)9-s + (1.68 − 2.91i)10-s + (0.254 + 0.948i)11-s − 0.516·12-s + (1.60 + 3.22i)13-s + (−0.764 + 4.36i)14-s + (−1.23 + 0.331i)15-s + (−2.48 + 4.29i)16-s + (−2.99 − 5.18i)17-s + ⋯ |
L(s) = 1 | + (1.14 − 0.306i)2-s + (−0.318 − 0.184i)3-s + (0.350 − 0.202i)4-s + (0.635 − 0.635i)5-s + (−0.421 − 0.112i)6-s + (−0.421 + 0.906i)7-s + (−0.498 + 0.498i)8-s + (−0.432 − 0.748i)9-s + (0.532 − 0.922i)10-s + (0.0766 + 0.285i)11-s − 0.148·12-s + (0.446 + 0.894i)13-s + (−0.204 + 1.16i)14-s + (−0.319 + 0.0856i)15-s + (−0.620 + 1.07i)16-s + (−0.725 − 1.25i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 + 0.452i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.891 + 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.41326 - 0.338331i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.41326 - 0.338331i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (1.11 - 2.39i)T \) |
| 13 | \( 1 + (-1.60 - 3.22i)T \) |
good | 2 | \( 1 + (-1.61 + 0.433i)T + (1.73 - i)T^{2} \) |
| 3 | \( 1 + (0.552 + 0.318i)T + (1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.42 + 1.42i)T - 5iT^{2} \) |
| 11 | \( 1 + (-0.254 - 0.948i)T + (-9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (2.99 + 5.18i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.71 + 0.726i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (-2.58 - 1.49i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.65 + 6.33i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-6.11 + 6.11i)T - 31iT^{2} \) |
| 37 | \( 1 + (1.24 + 4.63i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-0.886 - 3.30i)T + (-35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (0.748 - 0.432i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.17 - 2.17i)T + 47iT^{2} \) |
| 53 | \( 1 - 6.32T + 53T^{2} \) |
| 59 | \( 1 + (-0.131 + 0.491i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (11.2 - 6.51i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.827 - 0.221i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (3.01 - 11.2i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (-1.03 - 1.03i)T + 73iT^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 + (-1.23 + 1.23i)T - 83iT^{2} \) |
| 89 | \( 1 + (-7.75 + 2.07i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (-12.0 - 3.23i)T + (84.0 + 48.5i)T^{2} \) |
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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
−13.66419598267712506689226184388, −13.04971633676940068358197011865, −11.98054937736929134221887978707, −11.47999056927308768275449874484, −9.401481063778970580104470697180, −8.828315744963610966393478886177, −6.50673309191351787474509889798, −5.64974135306212417182863749447, −4.40694882426654236625056410290, −2.58932367142483130671681135039,
3.16181735494101965090229926164, 4.63298231516688399041894284081, 5.97234496732694591508256798935, 6.72179914874713216198860825091, 8.516910973916516328494831984785, 10.36815720799640763851039011863, 10.69959112048410040920961660643, 12.44978922368389111649386015227, 13.43870038572055605476832174822, 13.96109550287272233492910878324