L(s) = 1 | + (−0.809 + 0.587i)3-s + 4.41·7-s + (0.309 − 0.951i)9-s + (1.37 + 4.23i)11-s + (−1.77 + 5.46i)13-s + (−5.31 − 3.86i)17-s + (2.25 + 1.63i)19-s + (−3.57 + 2.59i)21-s + (0.406 + 1.25i)23-s + (0.309 + 0.951i)27-s + (3.91 − 2.84i)29-s + (0.159 + 0.115i)31-s + (−3.60 − 2.61i)33-s + (−2.53 + 7.80i)37-s + (−1.77 − 5.46i)39-s + ⋯ |
L(s) = 1 | + (−0.467 + 0.339i)3-s + 1.66·7-s + (0.103 − 0.317i)9-s + (0.414 + 1.27i)11-s + (−0.492 + 1.51i)13-s + (−1.28 − 0.936i)17-s + (0.516 + 0.375i)19-s + (−0.779 + 0.566i)21-s + (0.0847 + 0.260i)23-s + (0.0594 + 0.183i)27-s + (0.727 − 0.528i)29-s + (0.0286 + 0.0208i)31-s + (−0.626 − 0.455i)33-s + (−0.417 + 1.28i)37-s + (−0.284 − 0.875i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0658 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0658 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.567021925\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.567021925\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.809 - 0.587i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 4.41T + 7T^{2} \) |
| 11 | \( 1 + (-1.37 - 4.23i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (1.77 - 5.46i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (5.31 + 3.86i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-2.25 - 1.63i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-0.406 - 1.25i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-3.91 + 2.84i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.159 - 0.115i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (2.53 - 7.80i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-2.42 + 7.46i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 0.412T + 43T^{2} \) |
| 47 | \( 1 + (6.30 - 4.58i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-0.255 + 0.185i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-0.778 + 2.39i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-2.88 - 8.87i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (9.66 + 7.02i)T + (20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (0.411 - 0.299i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-4.87 - 14.9i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-2.77 + 2.01i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-4.34 - 3.15i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-3.50 - 10.7i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-4.10 + 2.98i)T + (29.9 - 92.2i)T^{2} \) |
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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
−9.581104318541599668860945087621, −9.049362394584956833675508759485, −8.047746276630053974214631751421, −7.13711988664403401390461858182, −6.61878962580731355574800039796, −5.22611557695102359752335461227, −4.62968469691148738752273257899, −4.17708779735743533489129680292, −2.34145136280636107842731881252, −1.48332945880184587298240228341,
0.69758099601283576992817580114, 1.86846887770454542165639910588, 3.12682458889775540161139356444, 4.42650373494478662279760911168, 5.19421463846287269636281801345, 5.90370639324886712049741925223, 6.85150462587396157670567332490, 7.894811662226993162189966555443, 8.296093381823924285666503438555, 9.069289071687913279254645259266