L(s) = 1 | + (0.309 − 0.951i)2-s + (−0.309 − 0.224i)3-s + (−0.809 − 0.587i)4-s + (−0.309 + 0.224i)6-s + 3·7-s + (−0.809 + 0.587i)8-s + (−0.881 − 2.71i)9-s + (1.30 − 4.02i)11-s + (0.118 + 0.363i)12-s + (−0.309 − 0.951i)13-s + (0.927 − 2.85i)14-s + (0.309 + 0.951i)16-s + (0.927 − 0.673i)17-s − 2.85·18-s + (−4.73 + 3.44i)19-s + ⋯ |
L(s) = 1 | + (0.218 − 0.672i)2-s + (−0.178 − 0.129i)3-s + (−0.404 − 0.293i)4-s + (−0.126 + 0.0916i)6-s + 1.13·7-s + (−0.286 + 0.207i)8-s + (−0.293 − 0.904i)9-s + (0.394 − 1.21i)11-s + (0.0340 + 0.104i)12-s + (−0.0857 − 0.263i)13-s + (0.247 − 0.762i)14-s + (0.0772 + 0.237i)16-s + (0.224 − 0.163i)17-s − 0.672·18-s + (−1.08 + 0.789i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0627 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0627 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.916963 - 0.976467i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.916963 - 0.976467i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.309 + 0.951i)T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (0.309 + 0.224i)T + (0.927 + 2.85i)T^{2} \) |
| 7 | \( 1 - 3T + 7T^{2} \) |
| 11 | \( 1 + (-1.30 + 4.02i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (0.309 + 0.951i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.927 + 0.673i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (4.73 - 3.44i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-0.545 + 1.67i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-7.66 - 5.56i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.190 + 0.138i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-2.57 - 7.91i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-0.454 - 1.40i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 6.23T + 43T^{2} \) |
| 47 | \( 1 + (9.66 + 7.02i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-8.47 - 6.15i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (1.38 + 4.25i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (2.73 - 8.42i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (8.28 - 6.01i)T + (20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (-2.42 - 1.76i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (2.38 - 7.33i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-5.85 - 4.25i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (3.66 - 2.66i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-1.38 + 4.25i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-7.73 - 5.62i)T + (29.9 + 92.2i)T^{2} \) |
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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
−11.77907674844595914551161986848, −11.04925375434233956829798195865, −10.16123943505611082537612023062, −8.785031845420394787484780371361, −8.241024897467997664671451892624, −6.56819403797208906871741567022, −5.56558711770748290358513296493, −4.30725658555648588292159298922, −3.02454084160291536161501268907, −1.18305760413442370997954833558,
2.16528768644006217321521727744, 4.38179267717941604464839415999, 4.91508447989423583463918343288, 6.24723276457481478104776501550, 7.44020280437899910002274364866, 8.181500419177666936464258977413, 9.261772523370093980111597644174, 10.47493034918356885708003902730, 11.38669587926124587099735520926, 12.32138838336799632444715142413