L(s) = 1 | + 0.517i·5-s − i·7-s + 3.18·11-s − 2.64·13-s + 5.15i·17-s + 8.23i·19-s − 4.38·23-s + 4.73·25-s − 5.11i·29-s − 0.548i·31-s + 0.517·35-s − 2.37·37-s + 0.473i·41-s + 0.816i·43-s − 0.493·47-s + ⋯ |
L(s) = 1 | + 0.231i·5-s − 0.377i·7-s + 0.959·11-s − 0.734·13-s + 1.25i·17-s + 1.88i·19-s − 0.913·23-s + 0.946·25-s − 0.949i·29-s − 0.0985i·31-s + 0.0874·35-s − 0.391·37-s + 0.0740i·41-s + 0.124i·43-s − 0.0719·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9944687180\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9944687180\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 - 0.517iT - 5T^{2} \) |
| 11 | \( 1 - 3.18T + 11T^{2} \) |
| 13 | \( 1 + 2.64T + 13T^{2} \) |
| 17 | \( 1 - 5.15iT - 17T^{2} \) |
| 19 | \( 1 - 8.23iT - 19T^{2} \) |
| 23 | \( 1 + 4.38T + 23T^{2} \) |
| 29 | \( 1 + 5.11iT - 29T^{2} \) |
| 31 | \( 1 + 0.548iT - 31T^{2} \) |
| 37 | \( 1 + 2.37T + 37T^{2} \) |
| 41 | \( 1 - 0.473iT - 41T^{2} \) |
| 43 | \( 1 - 0.816iT - 43T^{2} \) |
| 47 | \( 1 + 0.493T + 47T^{2} \) |
| 53 | \( 1 + 0.915iT - 53T^{2} \) |
| 59 | \( 1 + 12.7T + 59T^{2} \) |
| 61 | \( 1 + 3.40T + 61T^{2} \) |
| 67 | \( 1 + 0.585iT - 67T^{2} \) |
| 71 | \( 1 + 6.95T + 71T^{2} \) |
| 73 | \( 1 - 2.41T + 73T^{2} \) |
| 79 | \( 1 + 12.5iT - 79T^{2} \) |
| 83 | \( 1 - 0.215T + 83T^{2} \) |
| 89 | \( 1 - 10.1iT - 89T^{2} \) |
| 97 | \( 1 + 7.23T + 97T^{2} \) |
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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
−8.075835994863670625089700509141, −7.83888344628852576035045165483, −6.83087449480036116969828314817, −6.23329967564555751451860627398, −5.67263981756652379818938962916, −4.53553035385346188765849363220, −3.95357595790656772292047541855, −3.27352221134492759489483019173, −2.06183881410531933154728467418, −1.31546352526388467386483074985,
0.25175146107464040511601135566, 1.42435812566157622128542707853, 2.56033661841727328875630194291, 3.15218985572928627068303212052, 4.34904175259200534430264135394, 4.87682790080251092763931188584, 5.54219442348284233923990384354, 6.61357422680640655246730180021, 6.98382289223105395174382575527, 7.72019980159203098999856822992