L(s) = 1 | + 11.3i·2-s − 74.3i·3-s − 128.·4-s − 154.·5-s + 840.·6-s + 1.58e3·7-s − 1.44e3i·8-s + 1.03e3·9-s − 1.75e3i·10-s − 2.55e4·11-s + 9.51e3i·12-s + 1.03e4i·13-s + 1.79e4i·14-s + 1.15e4i·15-s + 1.63e4·16-s − 1.32e5·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.917i·3-s − 0.500·4-s − 0.247·5-s + 0.648·6-s + 0.660·7-s − 0.353i·8-s + 0.158·9-s − 0.175i·10-s − 1.74·11-s + 0.458i·12-s + 0.363i·13-s + 0.466i·14-s + 0.227i·15-s + 0.250·16-s − 1.59·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.100i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.994 + 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.00140793 - 0.0278446i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00140793 - 0.0278446i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 11.3iT \) |
| 19 | \( 1 + (-1.31e4 - 1.29e5i)T \) |
good | 3 | \( 1 + 74.3iT - 6.56e3T^{2} \) |
| 5 | \( 1 + 154.T + 3.90e5T^{2} \) |
| 7 | \( 1 - 1.58e3T + 5.76e6T^{2} \) |
| 11 | \( 1 + 2.55e4T + 2.14e8T^{2} \) |
| 13 | \( 1 - 1.03e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + 1.32e5T + 6.97e9T^{2} \) |
| 23 | \( 1 + 3.34e5T + 7.83e10T^{2} \) |
| 29 | \( 1 + 6.05e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 1.41e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 2.45e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 - 2.38e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 1.53e6T + 1.16e13T^{2} \) |
| 47 | \( 1 - 5.01e6T + 2.38e13T^{2} \) |
| 53 | \( 1 + 1.18e7iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 7.32e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 1.12e7T + 1.91e14T^{2} \) |
| 67 | \( 1 - 1.87e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 + 2.17e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 8.89e6T + 8.06e14T^{2} \) |
| 79 | \( 1 + 5.26e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 - 3.76e7T + 2.25e15T^{2} \) |
| 89 | \( 1 + 1.06e8iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 2.45e7iT - 7.83e15T^{2} \) |
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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
−15.28863488206357135272482517246, −13.80351887138484681447300093701, −13.06361283524679331584607977062, −11.73721168313566563973454366111, −10.15714306062328064874792756613, −8.232067821104726553083702082416, −7.57916230738888114686447742825, −6.12899648117112428628130348924, −4.50970950348286816079192229791, −2.00496441871339651760443164403,
0.01041672211631904907691093810, 2.35160751052819013078876328531, 4.13863697752549449108337185977, 5.22103694578882533572449738822, 7.70882392148477589659429834414, 9.107707825687351930297552891746, 10.48740750749065009338306923581, 11.07157262868562272452286266695, 12.66447548852752119158182574305, 13.78278142868192014201991462028