L(s) = 1 | + (0.575 − 1.91i)2-s + (2.61 − 4.52i)3-s + (−3.33 − 2.20i)4-s + (5.42 − 3.13i)5-s + (−7.16 − 7.60i)6-s + (−6.14 + 5.12i)8-s + (−9.14 − 15.8i)9-s + (−2.87 − 12.2i)10-s + (−4.90 + 8.49i)11-s + (−18.6 + 9.33i)12-s + 2.41i·13-s − 32.7i·15-s + (6.27 + 14.7i)16-s + (−3.44 + 5.97i)17-s + (−35.5 + 8.39i)18-s + (−1.38 − 2.40i)19-s + ⋯ |
L(s) = 1 | + (0.287 − 0.957i)2-s + (0.870 − 1.50i)3-s + (−0.834 − 0.551i)4-s + (1.08 − 0.626i)5-s + (−1.19 − 1.26i)6-s + (−0.768 + 0.640i)8-s + (−1.01 − 1.75i)9-s + (−0.287 − 1.22i)10-s + (−0.445 + 0.772i)11-s + (−1.55 + 0.778i)12-s + 0.185i·13-s − 2.18i·15-s + (0.392 + 0.919i)16-s + (−0.202 + 0.351i)17-s + (−1.97 + 0.466i)18-s + (−0.0730 − 0.126i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 - 0.223i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.974 - 0.223i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.295784 + 2.61320i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.295784 + 2.61320i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.575 + 1.91i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-2.61 + 4.52i)T + (-4.5 - 7.79i)T^{2} \) |
| 5 | \( 1 + (-5.42 + 3.13i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (4.90 - 8.49i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 2.41iT - 169T^{2} \) |
| 17 | \( 1 + (3.44 - 5.97i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (1.38 + 2.40i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-37.0 + 21.4i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 37.3iT - 841T^{2} \) |
| 31 | \( 1 + (-6.20 - 3.58i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (0.175 - 0.101i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 63.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + 35.3T + 1.84e3T^{2} \) |
| 47 | \( 1 + (32.8 - 18.9i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-47.3 - 27.3i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (52.3 - 90.7i)T + (-1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-37.9 + 21.8i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (15.5 - 26.9i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 23.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-34.6 + 59.9i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-17.2 + 9.96i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 5.11T + 6.88e3T^{2} \) |
| 89 | \( 1 + (-8.99 - 15.5i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 12.4T + 9.40e3T^{2} \) |
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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
−10.61932620773892524016625740322, −9.512175766002897591612488981328, −8.933166747686994985502246286270, −8.006575438602918576510079185031, −6.76941863927971661704616976621, −5.76421143903424183423864824873, −4.52686789810590765769407005735, −2.78221343405185663235303303545, −2.04730565234070811878037493094, −1.00762841990611218613985125347,
2.79678293683920947784927684904, 3.53753289692546965693128896794, 4.93791003588768022444092933030, 5.57225360229389068229942465369, 6.79630530481704419757161934970, 8.026352627133709000316675669230, 8.970896931304019848043602764602, 9.522511058640227168967865734338, 10.38447423134862406834119468203, 11.19737264415948015549665935990