| L(s) = 1 | + (1.10 + 0.883i)2-s + (−2.21 − 1.93i)3-s + (0.438 + 1.95i)4-s + (1.15 − 0.376i)5-s + (−0.737 − 4.10i)6-s + (−2.62 − 0.724i)7-s + (−1.23 + 2.54i)8-s + (0.762 + 5.63i)9-s + (1.61 + 0.608i)10-s + (−4.89 − 2.63i)11-s + (2.80 − 5.18i)12-s + (0.256 + 0.476i)13-s + (−2.25 − 3.12i)14-s + (−3.30 − 1.41i)15-s + (−3.61 + 1.71i)16-s + (−1.98 + 2.73i)17-s + ⋯ |
| L(s) = 1 | + (0.780 + 0.624i)2-s + (−1.28 − 1.11i)3-s + (0.219 + 0.975i)4-s + (0.518 − 0.168i)5-s + (−0.301 − 1.67i)6-s + (−0.992 − 0.273i)7-s + (−0.438 + 0.898i)8-s + (0.254 + 1.87i)9-s + (0.509 + 0.192i)10-s + (−1.47 − 0.794i)11-s + (0.811 − 1.49i)12-s + (0.0711 + 0.132i)13-s + (−0.603 − 0.834i)14-s + (−0.852 − 0.364i)15-s + (−0.903 + 0.427i)16-s + (−0.482 + 0.663i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0357i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0357i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.000674875 - 0.0377211i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.000674875 - 0.0377211i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.10 - 0.883i)T \) |
| 71 | \( 1 + (-7.48 + 3.86i)T \) |
| good | 3 | \( 1 + (2.21 + 1.93i)T + (0.402 + 2.97i)T^{2} \) |
| 5 | \( 1 + (-1.15 + 0.376i)T + (4.04 - 2.93i)T^{2} \) |
| 7 | \( 1 + (2.62 + 0.724i)T + (6.00 + 3.59i)T^{2} \) |
| 11 | \( 1 + (4.89 + 2.63i)T + (6.05 + 9.18i)T^{2} \) |
| 13 | \( 1 + (-0.256 - 0.476i)T + (-7.16 + 10.8i)T^{2} \) |
| 17 | \( 1 + (1.98 - 2.73i)T + (-5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-0.995 - 2.32i)T + (-13.1 + 13.7i)T^{2} \) |
| 23 | \( 1 + (-1.14 - 1.43i)T + (-5.11 + 22.4i)T^{2} \) |
| 29 | \( 1 + (3.85 + 6.45i)T + (-13.7 + 25.5i)T^{2} \) |
| 31 | \( 1 + (7.31 - 1.32i)T + (29.0 - 10.8i)T^{2} \) |
| 37 | \( 1 + (5.32 + 4.24i)T + (8.23 + 36.0i)T^{2} \) |
| 41 | \( 1 + (-2.25 + 4.67i)T + (-25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (-10.0 - 3.75i)T + (32.3 + 28.2i)T^{2} \) |
| 47 | \( 1 + (5.64 - 4.93i)T + (6.30 - 46.5i)T^{2} \) |
| 53 | \( 1 + (5.91 - 0.532i)T + (52.1 - 9.46i)T^{2} \) |
| 59 | \( 1 + (1.13 - 1.08i)T + (2.64 - 58.9i)T^{2} \) |
| 61 | \( 1 + (-7.84 + 2.16i)T + (52.3 - 31.2i)T^{2} \) |
| 67 | \( 1 + (-0.533 + 5.92i)T + (-65.9 - 11.9i)T^{2} \) |
| 73 | \( 1 + (-0.00957 + 0.213i)T + (-72.7 - 6.54i)T^{2} \) |
| 79 | \( 1 + (-6.35 - 0.860i)T + (76.1 + 21.0i)T^{2} \) |
| 83 | \( 1 + (-9.29 - 9.72i)T + (-3.72 + 82.9i)T^{2} \) |
| 89 | \( 1 + (-2.79 + 0.251i)T + (87.5 - 15.8i)T^{2} \) |
| 97 | \( 1 + (-4.05 - 8.41i)T + (-60.4 + 75.8i)T^{2} \) |
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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
−11.21963388545128359645411026278, −10.78037794004514198897466934988, −9.410167772967287435347622902105, −7.998442364394165385912820022544, −7.38012559363795867191421504487, −6.37053847826921997262190750469, −5.80649262820128501741492315646, −5.24096754568720608324506989384, −3.64018395974718199305427588548, −2.13418603739195620764428835935,
0.01755713557767204490581836496, 2.46088449246025802762368255814, 3.61657942469660074369996679144, 4.89481351183592175174038541475, 5.32710932720437863111139991108, 6.22289596406779480475096652387, 7.11090174952925168541931400417, 9.255158679243819274449796555977, 9.758866876564487395124553787920, 10.50568420963031824327927238166
