L(s) = 1 | + (−0.968 − 1.03i)2-s + (0.609 − 1.62i)3-s + (−0.124 + 1.99i)4-s + (−2.24 − 1.29i)5-s + (−2.26 + 0.942i)6-s + (−2.53 + 0.751i)7-s + (2.17 − 1.80i)8-s + (−2.25 − 1.97i)9-s + (0.839 + 3.57i)10-s + (−1.63 − 2.83i)11-s + (3.16 + 1.41i)12-s + 0.912·13-s + (3.23 + 1.88i)14-s + (−3.47 + 2.85i)15-s + (−3.96 − 0.495i)16-s + (2.39 + 4.14i)17-s + ⋯ |
L(s) = 1 | + (−0.684 − 0.728i)2-s + (0.351 − 0.936i)3-s + (−0.0621 + 0.998i)4-s + (−1.00 − 0.580i)5-s + (−0.923 + 0.384i)6-s + (−0.958 + 0.284i)7-s + (0.769 − 0.638i)8-s + (−0.752 − 0.658i)9-s + (0.265 + 1.13i)10-s + (−0.493 − 0.855i)11-s + (0.912 + 0.409i)12-s + 0.253·13-s + (0.863 + 0.504i)14-s + (−0.897 + 0.737i)15-s + (−0.992 − 0.123i)16-s + (0.580 + 1.00i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0295i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0295i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00767323 + 0.519742i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00767323 + 0.519742i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.968 + 1.03i)T \) |
| 3 | \( 1 + (-0.609 + 1.62i)T \) |
| 7 | \( 1 + (2.53 - 0.751i)T \) |
good | 5 | \( 1 + (2.24 + 1.29i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.63 + 2.83i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 0.912T + 13T^{2} \) |
| 17 | \( 1 + (-2.39 - 4.14i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.66 + 4.61i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.45 + 2.57i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 1.35T + 29T^{2} \) |
| 31 | \( 1 + (-8.18 + 4.72i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.59 - 0.922i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 5.91T + 41T^{2} \) |
| 43 | \( 1 + 8.00iT - 43T^{2} \) |
| 47 | \( 1 + (-3.29 + 5.70i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.841 + 1.45i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.50 - 0.867i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.72 - 8.18i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (10.8 - 6.27i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 0.603iT - 71T^{2} \) |
| 73 | \( 1 + (1.29 - 0.746i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.0625 - 0.108i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 0.246iT - 83T^{2} \) |
| 89 | \( 1 + (-1.80 + 3.12i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 5.10iT - 97T^{2} \) |
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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
−12.17128265866954224421942264232, −11.60582638270175162384257258722, −10.29035382156729841740754695600, −8.976150590441330350298765864167, −8.306013004913460933829713950293, −7.48000927669790005647426045961, −6.08980106806273318857476733163, −3.88922517985971844057853617941, −2.73321168783006184182023261993, −0.57842973314417070171856797140,
3.14766815219254795048625904515, 4.50611963779466687572668018959, 5.96488288590776610188766765438, 7.40862361593905690072920143491, 7.960615595283113889373411249895, 9.469810484113361836723879798334, 9.986459614602283287437007590405, 10.92538667633716026298339851573, 12.08610656098316032087557223769, 13.74522388332238388717934878934