L(s) = 1 | − i·2-s + (−1.08 + 1.35i)3-s − 4-s + (1.77 − 3.07i)5-s + (1.35 + 1.08i)6-s + (2.63 + 0.201i)7-s + i·8-s + (−0.645 − 2.92i)9-s + (−3.07 − 1.77i)10-s + (2.61 − 1.51i)11-s + (1.08 − 1.35i)12-s + (−0.888 + 0.513i)13-s + (0.201 − 2.63i)14-s + (2.22 + 5.73i)15-s + 16-s + (−0.809 + 1.40i)17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (−0.626 + 0.779i)3-s − 0.5·4-s + (0.794 − 1.37i)5-s + (0.551 + 0.442i)6-s + (0.997 + 0.0762i)7-s + 0.353i·8-s + (−0.215 − 0.976i)9-s + (−0.972 − 0.561i)10-s + (0.789 − 0.455i)11-s + (0.313 − 0.389i)12-s + (−0.246 + 0.142i)13-s + (0.0539 − 0.705i)14-s + (0.574 + 1.48i)15-s + 0.250·16-s + (−0.196 + 0.339i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.900845 - 0.465906i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.900845 - 0.465906i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (1.08 - 1.35i)T \) |
| 7 | \( 1 + (-2.63 - 0.201i)T \) |
good | 5 | \( 1 + (-1.77 + 3.07i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.61 + 1.51i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.888 - 0.513i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (0.809 - 1.40i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (7.12 - 4.11i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.90 - 1.67i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.70 + 2.13i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 5.98iT - 31T^{2} \) |
| 37 | \( 1 + (-2.92 - 5.06i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.0472 - 0.0817i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.05 + 5.29i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 5.14T + 47T^{2} \) |
| 53 | \( 1 + (2.76 + 1.59i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 8.84T + 59T^{2} \) |
| 61 | \( 1 - 4.69iT - 61T^{2} \) |
| 67 | \( 1 + 0.375T + 67T^{2} \) |
| 71 | \( 1 - 13.9iT - 71T^{2} \) |
| 73 | \( 1 + (-1.13 - 0.655i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 0.924T + 79T^{2} \) |
| 83 | \( 1 + (-5.43 + 9.40i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (2.35 + 4.07i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-13.3 - 7.69i)T + (48.5 + 84.0i)T^{2} \) |
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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
−12.93294270039203719857961070784, −12.10854043456195814974518512577, −11.18542894472985576656024133694, −10.18378430634614354079767902926, −9.093203231390590879796908047925, −8.481743281550610131894952886533, −6.07118058658196596784788366148, −5.02101313287783859930425530226, −4.11900023849943591704680287651, −1.52723067066618501694481823347,
2.19737931407136871382519441899, 4.70553179126040464194364107822, 6.11826326850018767392095687558, 6.83673457466841323799126016092, 7.73802044906836558123383288527, 9.242578802925028920533407013865, 10.70718687708100796870761623957, 11.27238493348260712114890407503, 12.71202129380532576198141397334, 13.70228106350918483040836110392