| L(s) = 1 | + (−0.104 − 0.994i)2-s + (0.100 − 1.72i)3-s + (−0.978 + 0.207i)4-s + 2.97·5-s + (−1.73 + 0.0812i)6-s + (0.990 − 3.04i)7-s + (0.309 + 0.951i)8-s + (−2.97 − 0.346i)9-s + (−0.311 − 2.96i)10-s + (−3.53 + 0.752i)11-s + (0.261 + 1.71i)12-s + (1.13 + 0.824i)13-s + (−3.13 − 0.666i)14-s + (0.298 − 5.15i)15-s + (0.913 − 0.406i)16-s + (4.08 − 4.53i)17-s + ⋯ |
| L(s) = 1 | + (−0.0739 − 0.703i)2-s + (0.0577 − 0.998i)3-s + (−0.489 + 0.103i)4-s + 1.33·5-s + (−0.706 + 0.0331i)6-s + (0.374 − 1.15i)7-s + (0.109 + 0.336i)8-s + (−0.993 − 0.115i)9-s + (−0.0984 − 0.936i)10-s + (−1.06 + 0.226i)11-s + (0.0755 + 0.494i)12-s + (0.314 + 0.228i)13-s + (−0.837 − 0.178i)14-s + (0.0769 − 1.33i)15-s + (0.228 − 0.101i)16-s + (0.989 − 1.09i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.460966 - 1.52326i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.460966 - 1.52326i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.100 + 1.72i)T \) |
| 31 | \( 1 + (4.92 + 2.59i)T \) |
| good | 5 | \( 1 - 2.97T + 5T^{2} \) |
| 7 | \( 1 + (-0.990 + 3.04i)T + (-5.66 - 4.11i)T^{2} \) |
| 11 | \( 1 + (3.53 - 0.752i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (-1.13 - 0.824i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-4.08 + 4.53i)T + (-1.77 - 16.9i)T^{2} \) |
| 19 | \( 1 + (0.408 + 0.181i)T + (12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (-2.57 + 2.85i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (-0.523 - 4.97i)T + (-28.3 + 6.02i)T^{2} \) |
| 37 | \( 1 + (-0.215 - 0.373i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3.81 - 2.77i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (7.26 - 5.27i)T + (13.2 - 40.8i)T^{2} \) |
| 47 | \( 1 + (-9.67 + 4.30i)T + (31.4 - 34.9i)T^{2} \) |
| 53 | \( 1 + (-6.98 - 1.48i)T + (48.4 + 21.5i)T^{2} \) |
| 59 | \( 1 + (8.80 - 3.91i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (-1.04 + 1.80i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 + (-13.4 - 2.86i)T + (64.8 + 28.8i)T^{2} \) |
| 73 | \( 1 + (-6.34 - 7.05i)T + (-7.63 + 72.6i)T^{2} \) |
| 79 | \( 1 + (2.28 + 7.02i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-7.31 - 3.25i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + (1.63 - 5.02i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-5.09 - 5.65i)T + (-10.1 + 96.4i)T^{2} \) |
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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
−10.47913955290296378571163311922, −9.709667628723822934947282556162, −8.725534771644053697040749407917, −7.65847085710348880291256120253, −6.95573627988019961906497950740, −5.71160800949527648917164754244, −4.87253498625597690791796795813, −3.17086959001407235334574726811, −2.09637530728262400670650946913, −0.980741905987742802278221372684,
2.12751173018824652467200326576, 3.43900703922026387050813034315, 5.06519851958537261637039304952, 5.59173320861820893018089369094, 6.10148500132712645277779023752, 7.79293992360556498763884623052, 8.647174863449032605930012498622, 9.257066351761697342167735790851, 10.19626169665894962637945634460, 10.67461012232462525808376962722
