| L(s) = 1 | + 2.81·5-s + 0.554·7-s + 1.00·11-s + 3.81·17-s + 0.753·19-s + 7.32·23-s + 2.91·25-s − 2.95·29-s + 2.58·31-s + 1.56·35-s + 7.85·37-s − 9.13·41-s − 5.59·43-s + 8.82·47-s − 6.69·49-s + 11.6·53-s + 2.82·55-s − 3.12·59-s + 0.631·61-s − 8.14·67-s − 13.0·71-s − 0.652·73-s + 0.557·77-s + 12.8·79-s + 3.25·83-s + 10.7·85-s − 1.14·89-s + ⋯ |
| L(s) = 1 | + 1.25·5-s + 0.209·7-s + 0.302·11-s + 0.925·17-s + 0.172·19-s + 1.52·23-s + 0.582·25-s − 0.547·29-s + 0.463·31-s + 0.263·35-s + 1.29·37-s − 1.42·41-s − 0.852·43-s + 1.28·47-s − 0.956·49-s + 1.59·53-s + 0.380·55-s − 0.406·59-s + 0.0807·61-s − 0.995·67-s − 1.55·71-s − 0.0764·73-s + 0.0634·77-s + 1.44·79-s + 0.357·83-s + 1.16·85-s − 0.120·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.030055679\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.030055679\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 2.81T + 5T^{2} \) |
| 7 | \( 1 - 0.554T + 7T^{2} \) |
| 11 | \( 1 - 1.00T + 11T^{2} \) |
| 17 | \( 1 - 3.81T + 17T^{2} \) |
| 19 | \( 1 - 0.753T + 19T^{2} \) |
| 23 | \( 1 - 7.32T + 23T^{2} \) |
| 29 | \( 1 + 2.95T + 29T^{2} \) |
| 31 | \( 1 - 2.58T + 31T^{2} \) |
| 37 | \( 1 - 7.85T + 37T^{2} \) |
| 41 | \( 1 + 9.13T + 41T^{2} \) |
| 43 | \( 1 + 5.59T + 43T^{2} \) |
| 47 | \( 1 - 8.82T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 + 3.12T + 59T^{2} \) |
| 61 | \( 1 - 0.631T + 61T^{2} \) |
| 67 | \( 1 + 8.14T + 67T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 + 0.652T + 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 - 3.25T + 83T^{2} \) |
| 89 | \( 1 + 1.14T + 89T^{2} \) |
| 97 | \( 1 - 5.79T + 97T^{2} \) |
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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
−8.050273942513119440620057538334, −7.31436918293718237994110959955, −6.58559684243429587443366300813, −5.88639701810655391541305769819, −5.29953765015235047461950373211, −4.61353907121839835854140854943, −3.50688883750808603281403944704, −2.73399630673461945956934855155, −1.78454033168666041437237392684, −0.979533906637678338820769832140,
0.979533906637678338820769832140, 1.78454033168666041437237392684, 2.73399630673461945956934855155, 3.50688883750808603281403944704, 4.61353907121839835854140854943, 5.29953765015235047461950373211, 5.88639701810655391541305769819, 6.58559684243429587443366300813, 7.31436918293718237994110959955, 8.050273942513119440620057538334
