L(s) = 1 | + (0.262 − 1.38i)2-s + (−2.91 − 0.781i)3-s + (−1.86 − 0.729i)4-s + (0.745 − 0.199i)5-s + (−1.85 + 3.84i)6-s + (−1.50 + 2.39i)8-s + (5.28 + 3.05i)9-s + (−0.0820 − 1.08i)10-s + (−0.333 + 1.24i)11-s + (4.85 + 3.57i)12-s + (0.919 − 0.919i)13-s − 2.32·15-s + (2.93 + 2.71i)16-s + (3.95 + 6.85i)17-s + (5.63 − 6.54i)18-s + (−0.478 − 1.78i)19-s + ⋯ |
L(s) = 1 | + (0.185 − 0.982i)2-s + (−1.68 − 0.450i)3-s + (−0.931 − 0.364i)4-s + (0.333 − 0.0893i)5-s + (−0.755 + 1.57i)6-s + (−0.530 + 0.847i)8-s + (1.76 + 1.01i)9-s + (−0.0259 − 0.344i)10-s + (−0.100 + 0.374i)11-s + (1.40 + 1.03i)12-s + (0.254 − 0.254i)13-s − 0.601·15-s + (0.734 + 0.678i)16-s + (0.959 + 1.66i)17-s + (1.32 − 1.54i)18-s + (−0.109 − 0.409i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.300 + 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.300 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.692115 - 0.507658i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.692115 - 0.507658i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.262 + 1.38i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (2.91 + 0.781i)T + (2.59 + 1.5i)T^{2} \) |
| 5 | \( 1 + (-0.745 + 0.199i)T + (4.33 - 2.5i)T^{2} \) |
| 11 | \( 1 + (0.333 - 1.24i)T + (-9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (-0.919 + 0.919i)T - 13iT^{2} \) |
| 17 | \( 1 + (-3.95 - 6.85i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.478 + 1.78i)T + (-16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (-3.33 - 1.92i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-5.25 + 5.25i)T - 29iT^{2} \) |
| 31 | \( 1 + (2.44 + 4.23i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.28 - 0.343i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 - 2.84iT - 41T^{2} \) |
| 43 | \( 1 + (-0.585 - 0.585i)T + 43iT^{2} \) |
| 47 | \( 1 + (-2.86 + 4.95i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.54 + 9.51i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (2.39 - 8.93i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-1.71 - 6.38i)T + (-52.8 + 30.5i)T^{2} \) |
| 67 | \( 1 + (-5.94 - 1.59i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 1.99iT - 71T^{2} \) |
| 73 | \( 1 + (-6.69 + 3.86i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4.63 - 8.02i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.78 + 4.78i)T - 83iT^{2} \) |
| 89 | \( 1 + (-1.84 - 1.06i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 9.01T + 97T^{2} \) |
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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
−10.27223469428863710884378211740, −9.845881303951246518186407439207, −8.509671065222664807476149423291, −7.44192749180644954973961948099, −6.16949439288336510248810865950, −5.68278771351034273963874730811, −4.81246973019107103321816139729, −3.73978801076530126299655348462, −2.00630070460261099463326454831, −0.913371554138145586908841095614,
0.75375365525738410209878802268, 3.37029385933612182749870474899, 4.65634469173750829201774894163, 5.23937833309784082504392947448, 5.99744110727210463394661929125, 6.72963368347010615558993312048, 7.53206816489312844626990973878, 8.828073174162162860800796574553, 9.692380942669384479334584110704, 10.41178627630864870969509687981