L(s) = 1 | + (−0.866 + 0.5i)3-s + (−1.86 + 1.23i)5-s + (0.499 − 0.866i)9-s + (2 + 3.46i)11-s − 6i·13-s + (1 − 2i)15-s + (−1.73 + i)17-s + (−3 + 5.19i)19-s + (1.73 + i)23-s + (1.96 − 4.59i)25-s + 0.999i·27-s − 6·29-s + (−1 − 1.73i)31-s + (−3.46 − 1.99i)33-s + (3.46 + 2i)37-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.288i)3-s + (−0.834 + 0.550i)5-s + (0.166 − 0.288i)9-s + (0.603 + 1.04i)11-s − 1.66i·13-s + (0.258 − 0.516i)15-s + (−0.420 + 0.242i)17-s + (−0.688 + 1.19i)19-s + (0.361 + 0.208i)23-s + (0.392 − 0.919i)25-s + 0.192i·27-s − 1.11·29-s + (−0.179 − 0.311i)31-s + (−0.603 − 0.348i)33-s + (0.569 + 0.328i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.208 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.208 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3263849673\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3263849673\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 + (1.86 - 1.23i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + (-2 - 3.46i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 + (1.73 - i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3 - 5.19i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.73 - i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.46 - 2i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + (-3.46 - 2i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (5.19 - 3i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-7 + 12.1i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.46 + 2i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (8.66 - 5i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 16iT - 83T^{2} \) |
| 89 | \( 1 + (4 - 6.92i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−8.327196118351050535511076730747, −7.80768178365690582113013349810, −7.02901729563145657930168506280, −6.28184233568045228983144273209, −5.48459609624045875983809038329, −4.52195087385897837808316065985, −3.83224199457062723471694908881, −3.03023538235333097740385598135, −1.69542043482815639547967671937, −0.12894544989900455822454417214,
1.04572725777777401426611315205, 2.23241214561685318696293250650, 3.59317916050644098694125908520, 4.30620370740051592516843906661, 5.00645988828839004505332497590, 5.98596763636294739369667124980, 6.87490614460097429736568580225, 7.18835967184589441082881164571, 8.402302280847789926254912421722, 8.883669675133795452109075011222