| L(s) = 1 | − 3-s + 2·5-s + 2·7-s + 9-s + 6·11-s + 13-s − 2·15-s − 2·17-s + 6·19-s − 2·21-s − 25-s − 27-s + 6·29-s + 6·31-s − 6·33-s + 4·35-s − 2·37-s − 39-s − 10·41-s − 8·43-s + 2·45-s + 6·47-s − 3·49-s + 2·51-s − 6·53-s + 12·55-s − 6·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 0.755·7-s + 1/3·9-s + 1.80·11-s + 0.277·13-s − 0.516·15-s − 0.485·17-s + 1.37·19-s − 0.436·21-s − 1/5·25-s − 0.192·27-s + 1.11·29-s + 1.07·31-s − 1.04·33-s + 0.676·35-s − 0.328·37-s − 0.160·39-s − 1.56·41-s − 1.21·43-s + 0.298·45-s + 0.875·47-s − 3/7·49-s + 0.280·51-s − 0.824·53-s + 1.61·55-s − 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.326292767\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.326292767\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.912725461042851430820741656750, −8.325889107858086631488400837557, −7.14292692276436999867746733687, −6.54689080179251243427232913926, −5.87367055113060473110467137059, −5.02186790895111483262642856513, −4.29490991180095637494576477321, −3.21201628736852196442434977525, −1.79263846979031199135308644293, −1.14011382779211435862107847319,
1.14011382779211435862107847319, 1.79263846979031199135308644293, 3.21201628736852196442434977525, 4.29490991180095637494576477321, 5.02186790895111483262642856513, 5.87367055113060473110467137059, 6.54689080179251243427232913926, 7.14292692276436999867746733687, 8.325889107858086631488400837557, 8.912725461042851430820741656750
