L(s) = 1 | + (0.707 − 1.22i)2-s + (−0.999 − 1.73i)4-s + (−1.93 − 1.11i)5-s + (−1.41 + 2.23i)7-s − 2.82·8-s + (1.5 − 2.59i)9-s + (−2.73 + 1.58i)10-s + (−3.23 − 5.60i)11-s − 2.66i·13-s + (1.73 + 3.31i)14-s + (−2.00 + 3.46i)16-s + (−2.12 − 3.67i)18-s + (7.23 + 4.17i)19-s + 4.47i·20-s − 9.16·22-s + (−0.184 + 0.320i)23-s + ⋯ |
L(s) = 1 | + (0.499 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.866 − 0.499i)5-s + (−0.534 + 0.845i)7-s − 0.999·8-s + (0.5 − 0.866i)9-s + (−0.866 + 0.499i)10-s + (−0.976 − 1.69i)11-s − 0.738i·13-s + (0.464 + 0.885i)14-s + (−0.500 + 0.866i)16-s + (−0.499 − 0.866i)18-s + (1.66 + 0.958i)19-s + 0.999i·20-s − 1.95·22-s + (−0.0385 + 0.0667i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 + 0.349i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.937 + 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.187420 - 1.04027i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.187420 - 1.04027i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 1.22i)T \) |
| 5 | \( 1 + (1.93 + 1.11i)T \) |
| 7 | \( 1 + (1.41 - 2.23i)T \) |
good | 3 | \( 1 + (-1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (3.23 + 5.60i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 2.66iT - 13T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-7.23 - 4.17i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.184 - 0.320i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.76 + 8.25i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 4.59iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + (-7.93 - 4.57i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.39 + 7.61i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (5.47 - 3.16i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (-10.9 - 6.32i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−11.63404797337433985334425750803, −10.71656159828220059973027935748, −9.602817959375798273362658641226, −8.786465833549814785152150736485, −7.73779956015661794728973834213, −5.95451758198064560883747429558, −5.34117194586843391410390880484, −3.70916961231326489597934472969, −3.02913769597745818547456351169, −0.70102546280214667658795974500,
2.87740903830420743672945003882, 4.31186020691732548076200522895, 4.91768792184638798279449746822, 6.70785253673065625249832432521, 7.40649332098398370651571644441, 7.79750892305032731197576140119, 9.456786585429390398451721854471, 10.31954755706839218370105733054, 11.50177773917338873014426110896, 12.47496223769727976513454362056