L(s) = 1 | + (−0.142 + 0.989i)2-s + (0.415 + 0.909i)3-s + (−0.959 − 0.281i)4-s + (0.841 + 0.540i)5-s + (−0.959 + 0.281i)6-s + (0.415 − 0.909i)7-s + (0.415 − 0.909i)8-s + (−0.654 + 0.755i)9-s + (−0.654 + 0.755i)10-s + (−0.654 + 0.755i)11-s + (−0.142 − 0.989i)12-s + (−0.142 + 0.989i)13-s + (0.841 + 0.540i)14-s + (−0.142 + 0.989i)15-s + (0.841 + 0.540i)16-s + (−0.654 + 0.755i)17-s + ⋯ |
L(s) = 1 | + (−0.142 + 0.989i)2-s + (0.415 + 0.909i)3-s + (−0.959 − 0.281i)4-s + (0.841 + 0.540i)5-s + (−0.959 + 0.281i)6-s + (0.415 − 0.909i)7-s + (0.415 − 0.909i)8-s + (−0.654 + 0.755i)9-s + (−0.654 + 0.755i)10-s + (−0.654 + 0.755i)11-s + (−0.142 − 0.989i)12-s + (−0.142 + 0.989i)13-s + (0.841 + 0.540i)14-s + (−0.142 + 0.989i)15-s + (0.841 + 0.540i)16-s + (−0.654 + 0.755i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2867852231 + 1.177351152i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2867852231 + 1.177351152i\) |
\(L(1)\) |
\(\approx\) |
\(0.7197069585 + 0.8548338677i\) |
\(L(1)\) |
\(\approx\) |
\(0.7197069585 + 0.8548338677i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 199 | \( 1 \) |
good | 2 | \( 1 + (-0.142 + 0.989i)T \) |
| 3 | \( 1 + (0.415 + 0.909i)T \) |
| 5 | \( 1 + (0.841 + 0.540i)T \) |
| 7 | \( 1 + (0.415 - 0.909i)T \) |
| 11 | \( 1 + (-0.654 + 0.755i)T \) |
| 13 | \( 1 + (-0.142 + 0.989i)T \) |
| 17 | \( 1 + (-0.654 + 0.755i)T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + (-0.959 + 0.281i)T \) |
| 29 | \( 1 + (-0.142 - 0.989i)T \) |
| 31 | \( 1 + (0.415 - 0.909i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (-0.654 + 0.755i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (0.415 - 0.909i)T \) |
| 53 | \( 1 + (-0.142 - 0.989i)T \) |
| 59 | \( 1 + (-0.959 + 0.281i)T \) |
| 61 | \( 1 + (0.841 - 0.540i)T \) |
| 67 | \( 1 + (0.841 + 0.540i)T \) |
| 71 | \( 1 + (0.841 - 0.540i)T \) |
| 73 | \( 1 + (-0.959 + 0.281i)T \) |
| 79 | \( 1 + (0.841 + 0.540i)T \) |
| 83 | \( 1 + (-0.142 + 0.989i)T \) |
| 89 | \( 1 + (-0.142 - 0.989i)T \) |
| 97 | \( 1 + (0.415 - 0.909i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−26.57045796565711974779324150889, −25.490919293308973846680944765443, −24.66164202592614423933327664254, −23.86836633783299670867956814049, −22.43421500781274257398398742699, −21.64634793888655290534778525656, −20.55278385648077663593708438502, −20.07788828732371469807742405974, −18.71205639387235989900473728536, −18.09295619414521049602781501341, −17.53109332447425027618436051743, −15.93243988552975558193420694262, −14.30810093176592429701447901670, −13.61878530423315922233078504560, −12.7024182140661266474065663667, −11.9963158760124933555199396192, −10.76775306519989994023843846841, −9.420871943457484304333893773142, −8.65197024887698438385290408450, −7.784510854457659505169958810464, −5.87504730529765393799533949174, −5.05291426081244463496050874871, −3.01831979411871509323386540113, −2.28927863145751553742356674732, −0.99576459872401436335208643376,
2.07739978545851977617590186897, 3.91091119975625763412285474068, 4.77989991462198609287259926909, 5.99352385948400964023070259326, 7.23765556891576182474643965893, 8.19046974449786343163744704630, 9.65564163493132214437423837467, 9.9694139125927713155118727621, 11.1885612677874537800224090546, 13.31916719964140199814737593164, 13.94654467940508123295574552210, 14.76263218397908503025397264744, 15.63524932829510305894280659127, 16.76495695966973847846509864882, 17.45652852486170623930017197459, 18.41429448475835000571280108248, 19.72688266700824000832796484883, 20.82051059620421311246845505373, 21.78324744398313604462678584099, 22.57110253584090368740849054920, 23.623131447176632877821740620664, 24.64023664418296311097850598989, 25.76355187257771523346809428056, 26.40840113129493789678876316651, 26.71563378114026727763274797620