L(s) = 1 | − 0.863·5-s − i·7-s + 2.62i·11-s + 1.01i·13-s − 2.34i·17-s + 4.09·19-s + 6.23·23-s − 4.25·25-s + 1.57·29-s + 6.29i·31-s + 0.863i·35-s + 5.18i·37-s − 10.1i·41-s + 0.496·43-s − 5.24·47-s + ⋯ |
L(s) = 1 | − 0.386·5-s − 0.377i·7-s + 0.792i·11-s + 0.282i·13-s − 0.568i·17-s + 0.939·19-s + 1.29·23-s − 0.850·25-s + 0.292·29-s + 1.13i·31-s + 0.145i·35-s + 0.851i·37-s − 1.58i·41-s + 0.0756·43-s − 0.764·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.605347596\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.605347596\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.863T + 5T^{2} \) |
| 11 | \( 1 - 2.62iT - 11T^{2} \) |
| 13 | \( 1 - 1.01iT - 13T^{2} \) |
| 17 | \( 1 + 2.34iT - 17T^{2} \) |
| 19 | \( 1 - 4.09T + 19T^{2} \) |
| 23 | \( 1 - 6.23T + 23T^{2} \) |
| 29 | \( 1 - 1.57T + 29T^{2} \) |
| 31 | \( 1 - 6.29iT - 31T^{2} \) |
| 37 | \( 1 - 5.18iT - 37T^{2} \) |
| 41 | \( 1 + 10.1iT - 41T^{2} \) |
| 43 | \( 1 - 0.496T + 43T^{2} \) |
| 47 | \( 1 + 5.24T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 + 4.40iT - 59T^{2} \) |
| 61 | \( 1 - 9.59iT - 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 + 2.11T + 71T^{2} \) |
| 73 | \( 1 + 8.27T + 73T^{2} \) |
| 79 | \( 1 + 9.44iT - 79T^{2} \) |
| 83 | \( 1 - 6.62iT - 83T^{2} \) |
| 89 | \( 1 + 10.2iT - 89T^{2} \) |
| 97 | \( 1 - 15.1T + 97T^{2} \) |
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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
−8.078242503563630571797942120406, −7.28571851678463154606676292543, −7.03794160385096781636628122614, −6.11638856388923000879974991364, −5.03521019265531245900571257746, −4.73290157049701178718404843182, −3.67706499906643058600325243628, −3.04484440967910635725045144746, −1.92707584839521682211975556428, −0.896463657554551123535157626632,
0.51768619396182043250279480428, 1.64898697303894968547134158882, 2.85235815877923769658427559439, 3.40294674950971766678901633525, 4.31475533974354139115407133305, 5.16902005460139379245107281374, 5.84743102887587689220574414377, 6.47638401749633877445699455240, 7.39700216376085619001182394574, 8.031284144281494344934005676805