L(s) = 1 | + (−2.91 + 0.781i)3-s + (−0.745 − 0.199i)5-s + (2.51 − 0.830i)7-s + (5.28 − 3.05i)9-s + (0.333 + 1.24i)11-s + (−0.919 − 0.919i)13-s + 2.32·15-s + (−3.95 + 6.85i)17-s + (−0.478 + 1.78i)19-s + (−6.67 + 4.38i)21-s + (−3.33 + 1.92i)23-s + (−3.81 − 2.20i)25-s + (−6.63 + 6.63i)27-s + (5.25 + 5.25i)29-s + (−2.44 + 4.23i)31-s + ⋯ |
L(s) = 1 | + (−1.68 + 0.450i)3-s + (−0.333 − 0.0893i)5-s + (0.949 − 0.313i)7-s + (1.76 − 1.01i)9-s + (0.100 + 0.374i)11-s + (−0.254 − 0.254i)13-s + 0.601·15-s + (−0.959 + 1.66i)17-s + (−0.109 + 0.409i)19-s + (−1.45 + 0.956i)21-s + (−0.694 + 0.401i)23-s + (−0.762 − 0.440i)25-s + (−1.27 + 1.27i)27-s + (0.975 + 0.975i)29-s + (−0.439 + 0.761i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.477 - 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.477 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.264077 + 0.444295i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.264077 + 0.444295i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2.51 + 0.830i)T \) |
good | 3 | \( 1 + (2.91 - 0.781i)T + (2.59 - 1.5i)T^{2} \) |
| 5 | \( 1 + (0.745 + 0.199i)T + (4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (-0.333 - 1.24i)T + (-9.52 + 5.5i)T^{2} \) |
| 13 | \( 1 + (0.919 + 0.919i)T + 13iT^{2} \) |
| 17 | \( 1 + (3.95 - 6.85i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.478 - 1.78i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (3.33 - 1.92i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-5.25 - 5.25i)T + 29iT^{2} \) |
| 31 | \( 1 + (2.44 - 4.23i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.28 + 0.343i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 2.84iT - 41T^{2} \) |
| 43 | \( 1 + (0.585 - 0.585i)T - 43iT^{2} \) |
| 47 | \( 1 + (-2.86 - 4.95i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.54 - 9.51i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (2.39 + 8.93i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (1.71 - 6.38i)T + (-52.8 - 30.5i)T^{2} \) |
| 67 | \( 1 + (5.94 - 1.59i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 1.99iT - 71T^{2} \) |
| 73 | \( 1 + (6.69 + 3.86i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.63 - 8.02i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.78 - 4.78i)T + 83iT^{2} \) |
| 89 | \( 1 + (1.84 - 1.06i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 9.01T + 97T^{2} \) |
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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
−11.25897327110587231913192941818, −10.64167022264029695554658272491, −10.06382531871618497866277378365, −8.649917998551578336001762593168, −7.61891897095296846973372063928, −6.52533678341053489909815709564, −5.67225797961951581475278467465, −4.64386274826416955581388409581, −4.02455145833222296011158318850, −1.52911035971663054550531336161,
0.41475289487211639090305180077, 2.16279477807421255216366032722, 4.34953885489127148328544266870, 5.10079361132724127317720329537, 6.04301011458435217365807801333, 6.98741347037354335353422745193, 7.77399514098644885445733254265, 9.004839199327635834980407954750, 10.22383164821006011077584257005, 11.18123188735169466930243397523