L(s) = 1 | + (0.562 + 0.826i)2-s + (−0.283 − 0.958i)3-s + (−0.367 + 0.930i)4-s + (−0.952 + 0.304i)5-s + (0.633 − 0.773i)6-s + (−0.525 + 0.850i)7-s + (−0.975 + 0.219i)8-s + (−0.839 + 0.544i)9-s + (−0.787 − 0.616i)10-s + (−0.666 − 0.745i)11-s + (0.996 + 0.0883i)12-s + (0.0663 − 0.997i)13-s + (−0.999 + 0.0442i)14-s + (0.562 + 0.826i)15-s + (−0.730 − 0.683i)16-s + (0.699 + 0.714i)17-s + ⋯ |
L(s) = 1 | + (0.562 + 0.826i)2-s + (−0.283 − 0.958i)3-s + (−0.367 + 0.930i)4-s + (−0.952 + 0.304i)5-s + (0.633 − 0.773i)6-s + (−0.525 + 0.850i)7-s + (−0.975 + 0.219i)8-s + (−0.839 + 0.544i)9-s + (−0.787 − 0.616i)10-s + (−0.666 − 0.745i)11-s + (0.996 + 0.0883i)12-s + (0.0663 − 0.997i)13-s + (−0.999 + 0.0442i)14-s + (0.562 + 0.826i)15-s + (−0.730 − 0.683i)16-s + (0.699 + 0.714i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8348563627 - 0.1798699459i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8348563627 - 0.1798699459i\) |
\(L(1)\) |
\(\approx\) |
\(0.8577450689 + 0.1522237621i\) |
\(L(1)\) |
\(\approx\) |
\(0.8577450689 + 0.1522237621i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 569 | \( 1 \) |
good | 2 | \( 1 + (0.562 + 0.826i)T \) |
| 3 | \( 1 + (-0.283 - 0.958i)T \) |
| 5 | \( 1 + (-0.952 + 0.304i)T \) |
| 7 | \( 1 + (-0.525 + 0.850i)T \) |
| 11 | \( 1 + (-0.666 - 0.745i)T \) |
| 13 | \( 1 + (0.0663 - 0.997i)T \) |
| 17 | \( 1 + (0.699 + 0.714i)T \) |
| 19 | \( 1 + (-0.367 - 0.930i)T \) |
| 23 | \( 1 + (0.964 - 0.262i)T \) |
| 29 | \( 1 + (0.699 - 0.714i)T \) |
| 31 | \( 1 + (0.903 + 0.428i)T \) |
| 37 | \( 1 + (-0.525 + 0.850i)T \) |
| 41 | \( 1 + (0.964 - 0.262i)T \) |
| 43 | \( 1 + (0.562 - 0.826i)T \) |
| 47 | \( 1 + (-0.921 + 0.387i)T \) |
| 53 | \( 1 + (0.633 - 0.773i)T \) |
| 59 | \( 1 + (-0.921 + 0.387i)T \) |
| 61 | \( 1 + (-0.598 + 0.801i)T \) |
| 67 | \( 1 + (0.699 - 0.714i)T \) |
| 71 | \( 1 + (-0.975 - 0.219i)T \) |
| 73 | \( 1 + (-0.367 - 0.930i)T \) |
| 79 | \( 1 + (-0.197 + 0.980i)T \) |
| 83 | \( 1 + (-0.448 - 0.894i)T \) |
| 89 | \( 1 + (0.903 - 0.428i)T \) |
| 97 | \( 1 + (0.814 - 0.580i)T \) |
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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
−23.13486120929955243927581144496, −22.77797746778906205834601648539, −21.39539089778729556841968563818, −20.902010636310134994008801377253, −20.19736578316266790501982772118, −19.43171610615009478167242293898, −18.62494341329345526149533682418, −17.33601902593879442434665091102, −16.31369183224474387848881098792, −15.806618113179794289373755645885, −14.762786417983139345026562967912, −14.080965521081308519902561627629, −12.86684338215199827438821198786, −12.136807879401795264238048669258, −11.30468450521330688581742093809, −10.51431445229884658199514651741, −9.768968509836076194726103308004, −8.94623151090929381637620472628, −7.56366703491740456367309559042, −6.40295798161383737813886535586, −5.091698916016540969157923302, −4.44173767869397981644059080280, −3.72022878064467550147465545876, −2.82211863999298998369682773959, −1.03233966435967264536709998617,
0.47088912496857656819734840130, 2.78566051163806612359518605721, 3.15336838312815700343938864112, 4.765388858090344414262304035956, 5.73364638767516137152339033686, 6.42121414898534835062730423362, 7.37061065058340399304021797508, 8.231521421776928096158350001167, 8.6922382113604918767276489348, 10.53295252665255960902061543810, 11.55921138190704569460336177827, 12.37939041472016095171713282314, 12.92898351815327491737315113444, 13.77530123481574988790790828357, 14.96303711095329409848332553821, 15.510434541750515231569837504028, 16.32054821507799115921832146564, 17.31992447280960500795246880072, 18.146096676450652464775404207891, 19.018070651879041120554234549466, 19.47302711043646361150689260365, 20.93399026683708830952202120076, 21.85994287211284097304806664784, 22.86964202675053411093521603893, 23.05944272230502408864087092726