L(s) = 1 | − 0.571i·2-s − 0.428·3-s + 1.67·4-s + i·5-s + 0.244i·6-s − 2.67i·7-s − 2.10i·8-s − 2.81·9-s + 0.571·10-s + 5.10i·11-s − 0.715·12-s + (−1.57 + 3.24i)13-s − 1.52·14-s − 0.428i·15-s + 2.14·16-s − 5.34·17-s + ⋯ |
L(s) = 1 | − 0.404i·2-s − 0.247·3-s + 0.836·4-s + 0.447i·5-s + 0.0999i·6-s − 1.01i·7-s − 0.742i·8-s − 0.938·9-s + 0.180·10-s + 1.53i·11-s − 0.206·12-s + (−0.435 + 0.899i)13-s − 0.408·14-s − 0.110i·15-s + 0.535·16-s − 1.29·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.899 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.905227 - 0.207727i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.905227 - 0.207727i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - iT \) |
| 13 | \( 1 + (1.57 - 3.24i)T \) |
good | 2 | \( 1 + 0.571iT - 2T^{2} \) |
| 3 | \( 1 + 0.428T + 3T^{2} \) |
| 7 | \( 1 + 2.67iT - 7T^{2} \) |
| 11 | \( 1 - 5.10iT - 11T^{2} \) |
| 17 | \( 1 + 5.34T + 17T^{2} \) |
| 19 | \( 1 + 6.24iT - 19T^{2} \) |
| 23 | \( 1 - 2.42T + 23T^{2} \) |
| 29 | \( 1 - 2.67T + 29T^{2} \) |
| 31 | \( 1 - 0.244iT - 31T^{2} \) |
| 37 | \( 1 - 3.32iT - 37T^{2} \) |
| 41 | \( 1 + 6.48iT - 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 - 2.67iT - 47T^{2} \) |
| 53 | \( 1 + 4.20T + 53T^{2} \) |
| 59 | \( 1 + 0.899iT - 59T^{2} \) |
| 61 | \( 1 + 5.81T + 61T^{2} \) |
| 67 | \( 1 + 2.18iT - 67T^{2} \) |
| 71 | \( 1 + 6.24iT - 71T^{2} \) |
| 73 | \( 1 - 10.9iT - 73T^{2} \) |
| 79 | \( 1 + 3.63T + 79T^{2} \) |
| 83 | \( 1 + 9.81iT - 83T^{2} \) |
| 89 | \( 1 - 7.63iT - 89T^{2} \) |
| 97 | \( 1 - 11.3iT - 97T^{2} \) |
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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
−14.88002925724217395648853301900, −13.71351142861903819214329729372, −12.37782784925945569524643485038, −11.29787395438233315157899428719, −10.63542131962754570323670551521, −9.319945487646810336757114859379, −7.26603661263541042190303201899, −6.65910692770411291565565913649, −4.48298773849430596679237318785, −2.46972928728870805519330112331,
2.80712830311512006685651363026, 5.51102560512337133794403396456, 6.14585082942880469777021637917, 8.035550335690015358908252363190, 8.850415962085183974280220793406, 10.75746894082536747495287934991, 11.60837775717027520466664113516, 12.59743636928104245143390027488, 14.10342648637081388964377878437, 15.16460029849831355120241114897