| L(s) = 1 | + (1.40 + 0.125i)2-s + (0.0214 + 0.142i)3-s + (1.96 + 0.354i)4-s + (−1.13 − 0.446i)5-s + (0.0123 + 0.202i)6-s + (0.813 + 2.51i)7-s + (2.72 + 0.746i)8-s + (2.84 − 0.878i)9-s + (−1.54 − 0.772i)10-s + (0.161 − 0.524i)11-s + (−0.00815 + 0.287i)12-s + (−2.40 + 0.549i)13-s + (0.829 + 3.64i)14-s + (0.0390 − 0.171i)15-s + (3.74 + 1.39i)16-s + (1.10 − 0.750i)17-s + ⋯ |
| L(s) = 1 | + (0.996 + 0.0888i)2-s + (0.0123 + 0.0819i)3-s + (0.984 + 0.177i)4-s + (−0.509 − 0.199i)5-s + (0.00502 + 0.0827i)6-s + (0.307 + 0.951i)7-s + (0.964 + 0.263i)8-s + (0.949 − 0.292i)9-s + (−0.489 − 0.244i)10-s + (0.0487 − 0.158i)11-s + (−0.00235 + 0.0828i)12-s + (−0.668 + 0.152i)13-s + (0.221 + 0.975i)14-s + (0.0100 − 0.0442i)15-s + (0.937 + 0.348i)16-s + (0.266 − 0.182i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.389i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.920 - 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.43720 + 0.494792i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.43720 + 0.494792i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.40 - 0.125i)T \) |
| 7 | \( 1 + (-0.813 - 2.51i)T \) |
| good | 3 | \( 1 + (-0.0214 - 0.142i)T + (-2.86 + 0.884i)T^{2} \) |
| 5 | \( 1 + (1.13 + 0.446i)T + (3.66 + 3.40i)T^{2} \) |
| 11 | \( 1 + (-0.161 + 0.524i)T + (-9.08 - 6.19i)T^{2} \) |
| 13 | \( 1 + (2.40 - 0.549i)T + (11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (-1.10 + 0.750i)T + (6.21 - 15.8i)T^{2} \) |
| 19 | \( 1 + (0.491 - 0.283i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.26 - 0.860i)T + (8.40 + 21.4i)T^{2} \) |
| 29 | \( 1 + (-0.690 + 1.43i)T + (-18.0 - 22.6i)T^{2} \) |
| 31 | \( 1 + (-0.121 + 0.209i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.95 - 0.371i)T + (36.5 - 5.51i)T^{2} \) |
| 41 | \( 1 + (5.21 + 6.53i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (5.20 + 4.15i)T + (9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (2.83 - 2.62i)T + (3.51 - 46.8i)T^{2} \) |
| 53 | \( 1 + (4.79 + 0.359i)T + (52.4 + 7.89i)T^{2} \) |
| 59 | \( 1 + (-8.44 + 3.31i)T + (43.2 - 40.1i)T^{2} \) |
| 61 | \( 1 + (8.13 - 0.609i)T + (60.3 - 9.09i)T^{2} \) |
| 67 | \( 1 + (1.01 + 0.587i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.32 - 0.640i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (4.62 + 4.28i)T + (5.45 + 72.7i)T^{2} \) |
| 79 | \( 1 + (-0.739 - 1.28i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-14.6 - 3.35i)T + (74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (1.22 - 0.377i)T + (73.5 - 50.1i)T^{2} \) |
| 97 | \( 1 + 4.78T + 97T^{2} \) |
| 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
−11.88042328836186208745696038488, −10.64830135387469495583008565465, −9.632781643774896896399796988784, −8.418245103473262559471367846953, −7.47102982962464100832564008939, −6.52600679670914344343207856843, −5.35150865291794141107902184657, −4.51906681619094787727564033559, −3.40019632658665550315871341237, −1.95836185282441714023474240560,
1.62442728116979431838360465351, 3.30000689786195026256639690000, 4.32304564024671825694720823562, 5.09531441550073957506275511635, 6.63354594120830626881791880611, 7.31889218429361332264039755423, 8.031268539655141571811268716227, 9.851564799536971050846767640491, 10.49506031760530783495390791619, 11.36678231462213174469852436745
