L(s) = 1 | + (0.654 − 0.755i)3-s + (−2.40 + 1.54i)5-s + (−2.21 − 0.887i)7-s + (−0.142 − 0.989i)9-s + (0.0929 + 1.95i)11-s + (6.57 − 0.627i)13-s + (−0.407 + 2.83i)15-s + (−0.498 − 2.05i)17-s + (6.01 − 2.40i)19-s + (−2.12 + 1.09i)21-s + (5.97 + 1.15i)23-s + (1.32 − 2.90i)25-s + (−0.841 − 0.540i)27-s + (3.14 − 5.44i)29-s + (6.99 + 0.668i)31-s + ⋯ |
L(s) = 1 | + (0.378 − 0.436i)3-s + (−1.07 + 0.692i)5-s + (−0.837 − 0.335i)7-s + (−0.0474 − 0.329i)9-s + (0.0280 + 0.588i)11-s + (1.82 − 0.174i)13-s + (−0.105 + 0.731i)15-s + (−0.120 − 0.498i)17-s + (1.37 − 0.552i)19-s + (−0.462 + 0.238i)21-s + (1.24 + 0.239i)23-s + (0.265 − 0.581i)25-s + (−0.161 − 0.104i)27-s + (0.583 − 1.01i)29-s + (1.25 + 0.119i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.295i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 + 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.42120 - 0.215022i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.42120 - 0.215022i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.654 + 0.755i)T \) |
| 67 | \( 1 + (-2.37 + 7.83i)T \) |
good | 5 | \( 1 + (2.40 - 1.54i)T + (2.07 - 4.54i)T^{2} \) |
| 7 | \( 1 + (2.21 + 0.887i)T + (5.06 + 4.83i)T^{2} \) |
| 11 | \( 1 + (-0.0929 - 1.95i)T + (-10.9 + 1.04i)T^{2} \) |
| 13 | \( 1 + (-6.57 + 0.627i)T + (12.7 - 2.46i)T^{2} \) |
| 17 | \( 1 + (0.498 + 2.05i)T + (-15.1 + 7.78i)T^{2} \) |
| 19 | \( 1 + (-6.01 + 2.40i)T + (13.7 - 13.1i)T^{2} \) |
| 23 | \( 1 + (-5.97 - 1.15i)T + (21.3 + 8.54i)T^{2} \) |
| 29 | \( 1 + (-3.14 + 5.44i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-6.99 - 0.668i)T + (30.4 + 5.86i)T^{2} \) |
| 37 | \( 1 + (-0.506 - 0.877i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.25 - 2.14i)T + (1.95 - 40.9i)T^{2} \) |
| 43 | \( 1 + (4.07 + 1.19i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 + (-2.70 - 7.82i)T + (-36.9 + 29.0i)T^{2} \) |
| 53 | \( 1 + (5.19 - 1.52i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (0.533 + 1.16i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (0.212 - 4.45i)T + (-60.7 - 5.79i)T^{2} \) |
| 71 | \( 1 + (1.57 - 6.51i)T + (-63.1 - 32.5i)T^{2} \) |
| 73 | \( 1 + (0.0356 - 0.748i)T + (-72.6 - 6.93i)T^{2} \) |
| 79 | \( 1 + (3.92 + 5.51i)T + (-25.8 + 74.6i)T^{2} \) |
| 83 | \( 1 + (-9.09 - 4.68i)T + (48.1 + 67.6i)T^{2} \) |
| 89 | \( 1 + (9.03 + 10.4i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (5.11 + 8.85i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.20204109744326085311655492668, −9.348038718498379467297119309849, −8.394717399054206255278809271821, −7.54161237018569326520557276652, −6.89947012394651747574131865942, −6.15562082335854101547712000793, −4.61663446657243729272109379462, −3.44174671342398725414766774478, −2.97958892116633965256085797318, −0.965347509662727587899732814237,
1.06509766667199405971251628009, 3.20541691590064249677091908463, 3.64552977585511994234855673842, 4.78829315286601952495098914045, 5.87027093436103615724636334099, 6.82791067325373969844866689669, 8.106685982896289793726440798768, 8.581300883952541485348234463659, 9.227476373350521943278844113027, 10.29417537410824398891598873100