| L(s) = 1 | + (1.94 + 1.36i)2-s + (−0.583 + 1.63i)3-s + (1.25 + 3.43i)4-s + (0.278 − 2.21i)5-s + (−3.36 + 2.38i)6-s + (−1.02 − 2.19i)7-s + (−1.02 + 3.81i)8-s + (−2.32 − 1.90i)9-s + (3.57 − 3.94i)10-s + (0.00275 + 0.00327i)11-s + (−6.34 + 0.0362i)12-s + (0.792 + 1.13i)13-s + (1.00 − 5.68i)14-s + (3.45 + 1.74i)15-s + (−1.59 + 1.33i)16-s + (0.510 + 1.90i)17-s + ⋯ |
| L(s) = 1 | + (1.37 + 0.964i)2-s + (−0.336 + 0.941i)3-s + (0.626 + 1.71i)4-s + (0.124 − 0.992i)5-s + (−1.37 + 0.972i)6-s + (−0.387 − 0.830i)7-s + (−0.361 + 1.34i)8-s + (−0.773 − 0.633i)9-s + (1.12 − 1.24i)10-s + (0.000829 + 0.000988i)11-s + (−1.83 + 0.0104i)12-s + (0.219 + 0.313i)13-s + (0.267 − 1.51i)14-s + (0.892 + 0.451i)15-s + (−0.398 + 0.334i)16-s + (0.123 + 0.461i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0195 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0195 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.32245 + 1.29683i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.32245 + 1.29683i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.583 - 1.63i)T \) |
| 5 | \( 1 + (-0.278 + 2.21i)T \) |
| good | 2 | \( 1 + (-1.94 - 1.36i)T + (0.684 + 1.87i)T^{2} \) |
| 7 | \( 1 + (1.02 + 2.19i)T + (-4.49 + 5.36i)T^{2} \) |
| 11 | \( 1 + (-0.00275 - 0.00327i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.792 - 1.13i)T + (-4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (-0.510 - 1.90i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (6.69 - 3.86i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.70 - 3.12i)T + (14.7 + 17.6i)T^{2} \) |
| 29 | \( 1 + (1.74 + 9.87i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (5.62 - 2.04i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-1.68 + 0.451i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (2.95 + 0.520i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (0.628 - 7.18i)T + (-42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (-1.86 + 0.871i)T + (30.2 - 36.0i)T^{2} \) |
| 53 | \( 1 + (-1.25 - 1.25i)T + 53iT^{2} \) |
| 59 | \( 1 + (-0.763 - 0.640i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (6.39 + 2.32i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (7.45 - 5.21i)T + (22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (-8.32 - 4.80i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.77 + 0.744i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-6.95 + 1.22i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-5.84 + 8.34i)T + (-28.3 - 77.9i)T^{2} \) |
| 89 | \( 1 + (-3.03 - 5.26i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-12.5 - 1.09i)T + (95.5 + 16.8i)T^{2} \) |
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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
−13.46399680731908787758290110576, −12.85742644560002373329163984833, −11.78200514402473686745087276485, −10.49301660818975897834838416799, −9.248241954065818409112085267465, −7.969461298332122778391294827287, −6.48538341439590775515702684167, −5.57636412926539227265591697278, −4.43008372458112006815342775881, −3.74481884488142499888186020387,
2.23074030717754782086570515940, 3.17600830262475947955786023585, 5.10832764401389604570173585160, 6.15595436406418870207449374988, 7.04727055610860338552706100519, 8.895861887321425422101063729644, 10.71497905932692764750988231546, 11.06784462771039139693032944106, 12.24431285660092245605502529727, 12.86246563022531750227906094368
