| L(s) = 1 | + (0.796 − 1.16i)2-s + (0.0980 − 0.995i)3-s + (−0.730 − 1.86i)4-s + (−1.15 + 0.615i)5-s + (−1.08 − 0.907i)6-s + (−0.518 − 2.60i)7-s + (−2.75 − 0.628i)8-s + (−0.980 − 0.195i)9-s + (−0.197 + 1.83i)10-s + (−0.408 + 0.497i)11-s + (−1.92 + 0.544i)12-s + (0.814 − 1.52i)13-s + (−3.45 − 1.47i)14-s + (0.499 + 1.20i)15-s + (−2.93 + 2.72i)16-s + (−0.703 + 1.69i)17-s + ⋯ |
| L(s) = 1 | + (0.563 − 0.826i)2-s + (0.0565 − 0.574i)3-s + (−0.365 − 0.930i)4-s + (−0.514 + 0.275i)5-s + (−0.442 − 0.370i)6-s + (−0.196 − 0.985i)7-s + (−0.974 − 0.222i)8-s + (−0.326 − 0.0650i)9-s + (−0.0625 + 0.580i)10-s + (−0.123 + 0.149i)11-s + (−0.555 + 0.157i)12-s + (0.225 − 0.422i)13-s + (−0.924 − 0.393i)14-s + (0.128 + 0.311i)15-s + (−0.732 + 0.680i)16-s + (−0.170 + 0.412i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.179i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 + 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.120006 - 1.32717i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.120006 - 1.32717i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.796 + 1.16i)T \) |
| 3 | \( 1 + (-0.0980 + 0.995i)T \) |
| good | 5 | \( 1 + (1.15 - 0.615i)T + (2.77 - 4.15i)T^{2} \) |
| 7 | \( 1 + (0.518 + 2.60i)T + (-6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (0.408 - 0.497i)T + (-2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-0.814 + 1.52i)T + (-7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (0.703 - 1.69i)T + (-12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (1.26 + 4.16i)T + (-15.7 + 10.5i)T^{2} \) |
| 23 | \( 1 + (-2.71 + 1.81i)T + (8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (-5.02 + 4.12i)T + (5.65 - 28.4i)T^{2} \) |
| 31 | \( 1 + (-4.77 + 4.77i)T - 31iT^{2} \) |
| 37 | \( 1 + (2.18 + 0.663i)T + (30.7 + 20.5i)T^{2} \) |
| 41 | \( 1 + (-1.58 - 2.37i)T + (-15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (0.365 + 3.70i)T + (-42.1 + 8.38i)T^{2} \) |
| 47 | \( 1 + (-10.4 - 4.31i)T + (33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (3.95 + 3.24i)T + (10.3 + 51.9i)T^{2} \) |
| 59 | \( 1 + (-2.53 - 4.73i)T + (-32.7 + 49.0i)T^{2} \) |
| 61 | \( 1 + (5.83 + 0.575i)T + (59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (0.642 + 0.0633i)T + (65.7 + 13.0i)T^{2} \) |
| 71 | \( 1 + (-13.1 + 2.62i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (-2.05 + 10.3i)T + (-67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (-11.3 + 4.70i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (10.3 - 3.12i)T + (69.0 - 46.1i)T^{2} \) |
| 89 | \( 1 + (5.93 + 3.96i)T + (34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (3.73 - 3.73i)T - 97iT^{2} \) |
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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
−10.96336665194564749361899132392, −10.37898505395286030008116176765, −9.241385250024066177403877007853, −8.059608378692546286753218899297, −7.01910264388776858528365920807, −6.10752080393835104198186381927, −4.68663995848586001826570319455, −3.70333418738868345330686792584, −2.52197795747395437058885363235, −0.75127133801248818819854738068,
2.81781766490356538144253579817, 3.98081976848282290535106442908, 4.98496878978626849801023853788, 5.89101884723261938544972653207, 6.92636393649672719837058807050, 8.234350674391985533883178696267, 8.720422359091601324428923509838, 9.689201956683720434701770251115, 11.01685527660245772560954314280, 12.13122065033004500701995518804
