| L(s) = 1 | + (−0.104 + 0.994i)2-s + (1.72 − 0.154i)3-s + (−0.978 − 0.207i)4-s + 1.43·5-s + (−0.0266 + 1.73i)6-s + (0.320 + 0.987i)7-s + (0.309 − 0.951i)8-s + (2.95 − 0.533i)9-s + (−0.150 + 1.42i)10-s + (−0.454 − 0.0965i)11-s + (−1.71 − 0.207i)12-s + (−1.95 + 1.42i)13-s + (−1.01 + 0.215i)14-s + (2.48 − 0.222i)15-s + (0.913 + 0.406i)16-s + (4.00 + 4.45i)17-s + ⋯ |
| L(s) = 1 | + (−0.0739 + 0.703i)2-s + (0.996 − 0.0892i)3-s + (−0.489 − 0.103i)4-s + 0.642·5-s + (−0.0108 + 0.707i)6-s + (0.121 + 0.373i)7-s + (0.109 − 0.336i)8-s + (0.984 − 0.177i)9-s + (−0.0475 + 0.452i)10-s + (−0.136 − 0.0291i)11-s + (−0.496 − 0.0599i)12-s + (−0.542 + 0.394i)13-s + (−0.271 + 0.0576i)14-s + (0.640 − 0.0573i)15-s + (0.228 + 0.101i)16-s + (0.971 + 1.07i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.552 - 0.833i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.552 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.85476 + 0.996476i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.85476 + 0.996476i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.104 - 0.994i)T \) |
| 3 | \( 1 + (-1.72 + 0.154i)T \) |
| 31 | \( 1 + (-2.29 - 5.07i)T \) |
| good | 5 | \( 1 - 1.43T + 5T^{2} \) |
| 7 | \( 1 + (-0.320 - 0.987i)T + (-5.66 + 4.11i)T^{2} \) |
| 11 | \( 1 + (0.454 + 0.0965i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (1.95 - 1.42i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-4.00 - 4.45i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-5.35 + 2.38i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (4.59 + 5.10i)T + (-2.40 + 22.8i)T^{2} \) |
| 29 | \( 1 + (0.188 - 1.79i)T + (-28.3 - 6.02i)T^{2} \) |
| 37 | \( 1 + (-2.52 + 4.36i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (6.24 + 4.53i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-8.05 - 5.85i)T + (13.2 + 40.8i)T^{2} \) |
| 47 | \( 1 + (3.25 + 1.45i)T + (31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (12.4 - 2.64i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (4.31 + 1.92i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (2.79 + 4.83i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + 5.60T + 67T^{2} \) |
| 71 | \( 1 + (2.65 - 0.565i)T + (64.8 - 28.8i)T^{2} \) |
| 73 | \( 1 + (10.6 - 11.7i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (-4.23 + 13.0i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (1.39 - 0.620i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (-2.48 - 7.66i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-5.80 + 6.45i)T + (-10.1 - 96.4i)T^{2} \) |
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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
−10.49714666085619337126774107156, −9.763947857533109952586491823827, −9.089726944419947380864160486510, −8.185366992758315907514988954408, −7.48938258501824833760549247095, −6.43797661274613011128473945095, −5.47811655710544297900675721645, −4.33254798614144183436613896357, −3.02459819846005791750417157119, −1.70497449656072057603904543931,
1.40717231478473248401712738406, 2.66653814277539455950651091261, 3.57254516803707819581467214746, 4.79281685597555364671913468356, 5.86134099560274402360348089931, 7.64284200906138206406503417982, 7.75704860304194036396767389808, 9.251513846723966018095386679762, 9.826346220979912131642160639595, 10.20379286738475508779496393039
