L(s) = 1 | + (−0.155 + 0.268i)2-s + (−1.03 − 1.78i)3-s + (0.951 + 1.64i)4-s + (0.5 + 0.866i)5-s + 0.639·6-s + (1.06 + 1.83i)7-s − 1.21·8-s + (−0.625 + 1.08i)9-s − 0.310·10-s + 5.48·11-s + (1.96 − 3.39i)12-s + (−0.826 − 1.43i)13-s − 0.657·14-s + (1.03 − 1.78i)15-s + (−1.71 + 2.97i)16-s + (1.42 − 2.46i)17-s + ⋯ |
L(s) = 1 | + (−0.109 + 0.189i)2-s + (−0.595 − 1.03i)3-s + (0.475 + 0.824i)4-s + (0.223 + 0.387i)5-s + 0.261·6-s + (0.400 + 0.694i)7-s − 0.428·8-s + (−0.208 + 0.361i)9-s − 0.0981·10-s + 1.65·11-s + (0.566 − 0.981i)12-s + (−0.229 − 0.396i)13-s − 0.175·14-s + (0.266 − 0.461i)15-s + (−0.428 + 0.742i)16-s + (0.344 − 0.597i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.259i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.12327 + 0.148389i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.12327 + 0.148389i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + (2.41 - 5.58i)T \) |
good | 2 | \( 1 + (0.155 - 0.268i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (1.03 + 1.78i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + (-1.06 - 1.83i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 - 5.48T + 11T^{2} \) |
| 13 | \( 1 + (0.826 + 1.43i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.42 + 2.46i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.19 - 2.07i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 1.29T + 23T^{2} \) |
| 29 | \( 1 + 0.459T + 29T^{2} \) |
| 31 | \( 1 + 7.00T + 31T^{2} \) |
| 41 | \( 1 + (3.63 + 6.30i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + 4.39T + 43T^{2} \) |
| 47 | \( 1 + 0.310T + 47T^{2} \) |
| 53 | \( 1 + (-6.00 + 10.4i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.14 - 7.18i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.10 + 8.83i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.664 + 1.15i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.293 + 0.508i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 + (-1.41 - 2.45i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (5.91 - 10.2i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-8.40 + 14.5i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 18.5T + 97T^{2} \) |
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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
−12.22727892360312240857312443559, −11.98217125742242793634081288702, −11.13401617987251934545524998816, −9.499395389577307890779286938186, −8.394751528940216652260682818390, −7.22616743952558598523282542911, −6.63884955838643196243366614048, −5.55742530962537891250011509433, −3.50268332517667265698659000127, −1.82700343890040531540529183737,
1.47485728759692458017305214418, 3.93234909917639854572488259697, 4.96907037415115289276646352299, 6.05703275672667457704431400639, 7.20389077172651188661137735711, 9.047212366797795936077329056675, 9.705536567431877452660555734566, 10.67765370289685005900996428547, 11.29212735096118860040485244384, 12.18511478595230186620024437472