L(s) = 1 | + (−0.707 − 0.707i)2-s + (−0.707 − 0.707i)3-s + 1.00i·4-s + (−1.91 + 1.16i)5-s + 1.00i·6-s + (0.707 − 0.707i)8-s + 1.00i·9-s + (2.17 + 0.531i)10-s − 1.76·11-s + (0.707 − 0.707i)12-s + (2.71 + 2.71i)13-s + (2.17 + 0.531i)15-s − 1.00·16-s + (−1.57 + 1.57i)17-s + (0.707 − 0.707i)18-s − 1.77·19-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (−0.408 − 0.408i)3-s + 0.500i·4-s + (−0.854 + 0.518i)5-s + 0.408i·6-s + (0.250 − 0.250i)8-s + 0.333i·9-s + (0.686 + 0.168i)10-s − 0.532·11-s + (0.204 − 0.204i)12-s + (0.752 + 0.752i)13-s + (0.560 + 0.137i)15-s − 0.250·16-s + (−0.380 + 0.380i)17-s + (0.166 − 0.166i)18-s − 0.406·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 + 0.375i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.926 + 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2928960133\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2928960133\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (1.91 - 1.16i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 1.76T + 11T^{2} \) |
| 13 | \( 1 + (-2.71 - 2.71i)T + 13iT^{2} \) |
| 17 | \( 1 + (1.57 - 1.57i)T - 17iT^{2} \) |
| 19 | \( 1 + 1.77T + 19T^{2} \) |
| 23 | \( 1 + (-2.86 + 2.86i)T - 23iT^{2} \) |
| 29 | \( 1 + 3.84iT - 29T^{2} \) |
| 31 | \( 1 - 10.3iT - 31T^{2} \) |
| 37 | \( 1 + (2.35 + 2.35i)T + 37iT^{2} \) |
| 41 | \( 1 + 11.8iT - 41T^{2} \) |
| 43 | \( 1 + (-3.46 + 3.46i)T - 43iT^{2} \) |
| 47 | \( 1 + (4.34 - 4.34i)T - 47iT^{2} \) |
| 53 | \( 1 + (-0.290 + 0.290i)T - 53iT^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 + 6.78iT - 61T^{2} \) |
| 67 | \( 1 + (-5.40 - 5.40i)T + 67iT^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 + (7.51 + 7.51i)T + 73iT^{2} \) |
| 79 | \( 1 + 12.6iT - 79T^{2} \) |
| 83 | \( 1 + (1.94 + 1.94i)T + 83iT^{2} \) |
| 89 | \( 1 - 1.11T + 89T^{2} \) |
| 97 | \( 1 + (7.26 - 7.26i)T - 97iT^{2} \) |
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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
−8.918107818852466931718865098064, −8.504266880802407484891261198284, −7.51816570477694369275979640809, −6.89658795793149554977953427295, −6.10426382003146860031948965609, −4.77929572963140189828754046818, −3.89281305036276344537245398254, −2.87527209757900912935912685102, −1.71012556353767114708220344878, −0.17009405495697231199488439487,
1.13094861413961579070625612183, 2.97644110829572981386297376130, 4.11643863780062651470213467757, 4.96122164286932164904771576611, 5.72065482679258166955093331394, 6.65019754917406568527663370674, 7.65210931681433548864198253856, 8.184548349643937972740577274453, 9.000845462978010630825955686458, 9.716443113140875865340138272936