| L(s) = 1 | + (−1.04 + 0.137i)2-s + (1.09 + 2.22i)3-s + (−0.863 + 0.231i)4-s + (0.751 − 0.659i)5-s + (−1.45 − 2.17i)6-s + (−0.614 − 2.57i)7-s + (2.81 − 1.16i)8-s + (−1.92 + 2.50i)9-s + (−0.693 + 0.790i)10-s + (4.60 − 0.301i)11-s + (−1.46 − 1.66i)12-s + (0.0848 + 0.0848i)13-s + (0.993 + 2.59i)14-s + (2.29 + 0.949i)15-s + (−1.22 + 0.706i)16-s + (4.09 − 0.470i)17-s + ⋯ |
| L(s) = 1 | + (−0.737 + 0.0970i)2-s + (0.633 + 1.28i)3-s + (−0.431 + 0.115i)4-s + (0.336 − 0.294i)5-s + (−0.592 − 0.886i)6-s + (−0.232 − 0.972i)7-s + (0.994 − 0.411i)8-s + (−0.641 + 0.835i)9-s + (−0.219 + 0.250i)10-s + (1.38 − 0.0909i)11-s + (−0.422 − 0.481i)12-s + (0.0235 + 0.0235i)13-s + (0.265 + 0.694i)14-s + (0.592 + 0.245i)15-s + (−0.306 + 0.176i)16-s + (0.993 − 0.114i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 595 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.679 - 0.733i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 595 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.679 - 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.14415 + 0.499958i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.14415 + 0.499958i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.751 + 0.659i)T \) |
| 7 | \( 1 + (0.614 + 2.57i)T \) |
| 17 | \( 1 + (-4.09 + 0.470i)T \) |
| good | 2 | \( 1 + (1.04 - 0.137i)T + (1.93 - 0.517i)T^{2} \) |
| 3 | \( 1 + (-1.09 - 2.22i)T + (-1.82 + 2.38i)T^{2} \) |
| 11 | \( 1 + (-4.60 + 0.301i)T + (10.9 - 1.43i)T^{2} \) |
| 13 | \( 1 + (-0.0848 - 0.0848i)T + 13iT^{2} \) |
| 19 | \( 1 + (-0.114 - 0.873i)T + (-18.3 + 4.91i)T^{2} \) |
| 23 | \( 1 + (-1.19 - 0.589i)T + (14.0 + 18.2i)T^{2} \) |
| 29 | \( 1 + (1.12 + 5.64i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + (-1.29 + 0.640i)T + (18.8 - 24.5i)T^{2} \) |
| 37 | \( 1 + (0.456 - 6.96i)T + (-36.6 - 4.82i)T^{2} \) |
| 41 | \( 1 + (0.299 - 1.50i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (0.822 + 1.98i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (1.62 - 6.07i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-2.77 - 3.61i)T + (-13.7 + 51.1i)T^{2} \) |
| 59 | \( 1 + (9.13 + 1.20i)T + (56.9 + 15.2i)T^{2} \) |
| 61 | \( 1 + (2.46 + 7.25i)T + (-48.3 + 37.1i)T^{2} \) |
| 67 | \( 1 + (-14.0 - 8.13i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.69 - 1.79i)T + (27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (1.16 + 0.395i)T + (57.9 + 44.4i)T^{2} \) |
| 79 | \( 1 + (-2.86 + 5.80i)T + (-48.0 - 62.6i)T^{2} \) |
| 83 | \( 1 + (-2.47 + 5.97i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (-3.91 - 1.04i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-1.75 + 0.349i)T + (89.6 - 37.1i)T^{2} \) |
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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
−10.24502468023409981735654969370, −9.762109919897021269247660913738, −9.283696818991892491137078657068, −8.435751521675826661154644937331, −7.58897572814517649481086488810, −6.38119430886597446299174525475, −4.87584448608171859511856455071, −4.09360687116996782493253927748, −3.38597466122634722573616703733, −1.16793562599901806473363274295,
1.21238421144276323715047625441, 2.17966712330767489548265747179, 3.52553913976255782285011419450, 5.20551136534038824133538576684, 6.31940930104829895387685004849, 7.14690276782270384913921871484, 8.062873877056211084864382641047, 8.910840524979719811141448966062, 9.311586015950116693079599793920, 10.33089027910864416420084542462
