L(s) = 1 | − 1.56i·2-s − 0.438·4-s − 3.56i·5-s + 0.561i·7-s − 2.43i·8-s − 5.56·10-s + 2i·11-s + 0.876·14-s − 4.68·16-s − 1.56·17-s − 7.12i·19-s + 1.56i·20-s + 3.12·22-s + 2·23-s − 7.68·25-s + ⋯ |
L(s) = 1 | − 1.10i·2-s − 0.219·4-s − 1.59i·5-s + 0.212i·7-s − 0.862i·8-s − 1.75·10-s + 0.603i·11-s + 0.234·14-s − 1.17·16-s − 0.378·17-s − 1.63i·19-s + 0.349i·20-s + 0.665·22-s + 0.417·23-s − 1.53·25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.388933740\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.388933740\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 1.56iT - 2T^{2} \) |
| 5 | \( 1 + 3.56iT - 5T^{2} \) |
| 7 | \( 1 - 0.561iT - 7T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 17 | \( 1 + 1.56T + 17T^{2} \) |
| 19 | \( 1 + 7.12iT - 19T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 + 6.68T + 29T^{2} \) |
| 31 | \( 1 + 2.56iT - 31T^{2} \) |
| 37 | \( 1 - 7.56iT - 37T^{2} \) |
| 41 | \( 1 + 1.56iT - 41T^{2} \) |
| 43 | \( 1 + 4.56T + 43T^{2} \) |
| 47 | \( 1 + 8.24iT - 47T^{2} \) |
| 53 | \( 1 - 0.684T + 53T^{2} \) |
| 59 | \( 1 - 2.87iT - 59T^{2} \) |
| 61 | \( 1 - 3.87T + 61T^{2} \) |
| 67 | \( 1 + 4.56iT - 67T^{2} \) |
| 71 | \( 1 - 14iT - 71T^{2} \) |
| 73 | \( 1 + 10.1iT - 73T^{2} \) |
| 79 | \( 1 - 5.43T + 79T^{2} \) |
| 83 | \( 1 + 0.876iT - 83T^{2} \) |
| 89 | \( 1 + 4.87iT - 89T^{2} \) |
| 97 | \( 1 - 8.56iT - 97T^{2} \) |
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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
−9.142316132817521041099812793427, −8.589651936601402443002036017908, −7.43226082500921078022016950993, −6.60135736738156978987578188732, −5.32425178048999822647102106585, −4.67052590880792840773830734182, −3.83474350001262017664087974199, −2.56165624439918345731330092105, −1.64666864131275140311928411999, −0.52022721083876749509999785499,
2.00553202849314730377912392907, 3.09301942385086955850903871728, 3.98139402152287830611797281540, 5.43106179969596408526065099035, 6.06467410282436298617906518941, 6.73343977855231808573855731968, 7.42707564563321824271228809990, 7.975853888016240526665342666786, 8.938244882488699543498362840123, 9.966469536346010573688524610827