L(s) = 1 | + (−0.995 − 0.0950i)2-s + (0.235 − 0.971i)3-s + (0.981 + 0.189i)4-s + (−0.142 − 0.989i)5-s + (−0.327 + 0.945i)6-s + (0.723 − 0.690i)7-s + (−0.959 − 0.281i)8-s + (−0.888 − 0.458i)9-s + (0.0475 + 0.998i)10-s + (0.841 − 0.540i)11-s + (0.415 − 0.909i)12-s + (−0.995 − 0.0950i)13-s + (−0.786 + 0.618i)14-s + (−0.995 − 0.0950i)15-s + (0.928 + 0.371i)16-s + (0.841 − 0.540i)17-s + ⋯ |
L(s) = 1 | + (−0.995 − 0.0950i)2-s + (0.235 − 0.971i)3-s + (0.981 + 0.189i)4-s + (−0.142 − 0.989i)5-s + (−0.327 + 0.945i)6-s + (0.723 − 0.690i)7-s + (−0.959 − 0.281i)8-s + (−0.888 − 0.458i)9-s + (0.0475 + 0.998i)10-s + (0.841 − 0.540i)11-s + (0.415 − 0.909i)12-s + (−0.995 − 0.0950i)13-s + (−0.786 + 0.618i)14-s + (−0.995 − 0.0950i)15-s + (0.928 + 0.371i)16-s + (0.841 − 0.540i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.740 - 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.740 - 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2878628548 - 0.7450598816i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2878628548 - 0.7450598816i\) |
\(L(1)\) |
\(\approx\) |
\(0.6041167297 - 0.4842994165i\) |
\(L(1)\) |
\(\approx\) |
\(0.6041167297 - 0.4842994165i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 199 | \( 1 \) |
good | 2 | \( 1 + (-0.995 - 0.0950i)T \) |
| 3 | \( 1 + (0.235 - 0.971i)T \) |
| 5 | \( 1 + (-0.142 - 0.989i)T \) |
| 7 | \( 1 + (0.723 - 0.690i)T \) |
| 11 | \( 1 + (0.841 - 0.540i)T \) |
| 13 | \( 1 + (-0.995 - 0.0950i)T \) |
| 17 | \( 1 + (0.841 - 0.540i)T \) |
| 19 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + (0.981 - 0.189i)T \) |
| 29 | \( 1 + (0.580 + 0.814i)T \) |
| 31 | \( 1 + (0.235 + 0.971i)T \) |
| 37 | \( 1 + (-0.5 - 0.866i)T \) |
| 41 | \( 1 + (-0.888 - 0.458i)T \) |
| 43 | \( 1 + (-0.5 + 0.866i)T \) |
| 47 | \( 1 + (0.723 - 0.690i)T \) |
| 53 | \( 1 + (-0.995 + 0.0950i)T \) |
| 59 | \( 1 + (-0.654 - 0.755i)T \) |
| 61 | \( 1 + (-0.142 + 0.989i)T \) |
| 67 | \( 1 + (-0.142 - 0.989i)T \) |
| 71 | \( 1 + (-0.786 - 0.618i)T \) |
| 73 | \( 1 + (0.981 - 0.189i)T \) |
| 79 | \( 1 + (0.928 + 0.371i)T \) |
| 83 | \( 1 + (0.415 + 0.909i)T \) |
| 89 | \( 1 + (0.580 + 0.814i)T \) |
| 97 | \( 1 + (0.235 + 0.971i)T \) |
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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
−27.32089604040489735934250348966, −26.50700527133028648378569020758, −25.58214923220080298377216295941, −24.930270981939013690998261800155, −23.58560730089863426689899750202, −22.22552212738115772475805880256, −21.5558834015227341513406429475, −20.56482062214875477229938623473, −19.43234140058620731606674747674, −18.85378122283308760254855038553, −17.39927961993148020622694325637, −17.05284308561414684349074315265, −15.33177504625737442114052514003, −15.11590523298179474391317769494, −14.25993274427867428024999806955, −11.98946917972043102068418624535, −11.30450643197323378544195544350, −10.25329393510376841911976809020, −9.47280555432584445479314111190, −8.423008602744377133296575258127, −7.355143372876671014551916972352, −6.12213301874899644148773971341, −4.71304998757962143001621071616, −3.10284768835757761258478646323, −2.08184317236244072194792697793,
0.86391337544622682394943153681, 1.72666904527200255621277465747, 3.37739800524506565927794638184, 5.18333161787770764550495987425, 6.68136648524422305677546309806, 7.64734191647526343463403588722, 8.42711603131789028210678214154, 9.31402184538619137910682794323, 10.71576777534919564489334883800, 11.96597967741715402793643928695, 12.418194239095651439984968413685, 13.918040732400637556051208988045, 14.81800059437768902666964854302, 16.53155623416794934516025922704, 16.99869224050840389440246262076, 17.84879409787050782265975858543, 19.05184567443783191968374996693, 19.71593138735663232516618883777, 20.50311784249979169018573157107, 21.35799349984426282022220924271, 23.20507363407839410075275404405, 24.07056959720352797196632354935, 24.87960918469577899933132250109, 25.282929952872168732586422386233, 26.91768060886936383091045586882