L(s) = 1 | + (−0.780 − 1.35i)2-s + (0.0458 − 1.73i)3-s + (−0.219 + 0.380i)4-s + (−0.5 + 0.866i)5-s + (−2.37 + 1.29i)6-s + (−2.47 − 4.28i)7-s − 2.43·8-s + (−2.99 − 0.158i)9-s + 1.56·10-s + (0.149 + 0.259i)11-s + (0.648 + 0.397i)12-s + (0.5 − 0.866i)13-s + (−3.86 + 6.68i)14-s + (1.47 + 0.905i)15-s + (2.34 + 4.05i)16-s + 8.20·17-s + ⋯ |
L(s) = 1 | + (−0.552 − 0.956i)2-s + (0.0264 − 0.999i)3-s + (−0.109 + 0.190i)4-s + (−0.223 + 0.387i)5-s + (−0.970 + 0.526i)6-s + (−0.934 − 1.61i)7-s − 0.861·8-s + (−0.998 − 0.0529i)9-s + 0.493·10-s + (0.0451 + 0.0781i)11-s + (0.187 + 0.114i)12-s + (0.138 − 0.240i)13-s + (−1.03 + 1.78i)14-s + (0.381 + 0.233i)15-s + (0.585 + 1.01i)16-s + 1.98·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.121 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.121 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.347171 + 0.392175i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.347171 + 0.392175i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.0458 + 1.73i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 + (-0.5 + 0.866i)T \) |
good | 2 | \( 1 + (0.780 + 1.35i)T + (-1 + 1.73i)T^{2} \) |
| 7 | \( 1 + (2.47 + 4.28i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.149 - 0.259i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 - 8.20T + 17T^{2} \) |
| 19 | \( 1 + 3.11T + 19T^{2} \) |
| 23 | \( 1 + (0.916 - 1.58i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.905 + 1.56i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.503 - 0.872i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 4.66T + 37T^{2} \) |
| 41 | \( 1 + (0.897 - 1.55i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5.48 + 9.50i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.79 - 4.83i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 0.636T + 53T^{2} \) |
| 59 | \( 1 + (-0.367 + 0.636i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.66 - 2.88i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.07 + 8.78i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 0.479T + 71T^{2} \) |
| 73 | \( 1 - 8.42T + 73T^{2} \) |
| 79 | \( 1 + (7.32 + 12.6i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (8.26 + 14.3i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 13.5T + 89T^{2} \) |
| 97 | \( 1 + (7.11 + 12.3i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.28697984511199048700481570523, −9.488713928824121862066579823405, −8.242374894109151967391058751359, −7.39743016606893149426311332113, −6.64593578299912190950420348220, −5.71367164208300837008318677411, −3.70168905857336830161737861662, −3.04941616785079294350244831047, −1.50722512598878279567892070145, −0.35305248792954965504176638427,
2.76022643470392563807647022084, 3.67263799676139795956438436026, 5.32825395740056131523938999628, 5.80156421779545123797780104770, 6.78042160869149358072599641338, 8.249454592541841286314940428830, 8.544537517200186313478967576646, 9.511934292416570495262249859099, 9.915537462410246252906192591827, 11.36415326262798922345852246430