L(s) = 1 | − 3-s − 5-s − 3.67i·7-s + 9-s − 4.75i·11-s + 4.23i·13-s + 15-s + 1.85·17-s + (−1.32 + 4.15i)19-s + 3.67i·21-s + 4.55i·23-s + 25-s − 27-s + 4.50i·29-s − 2.95·31-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.38i·7-s + 0.333·9-s − 1.43i·11-s + 1.17i·13-s + 0.258·15-s + 0.450·17-s + (−0.302 + 0.952i)19-s + 0.801i·21-s + 0.950i·23-s + 0.200·25-s − 0.192·27-s + 0.836i·29-s − 0.531·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.214i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 - 0.214i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.149245487\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.149245487\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + (1.32 - 4.15i)T \) |
good | 7 | \( 1 + 3.67iT - 7T^{2} \) |
| 11 | \( 1 + 4.75iT - 11T^{2} \) |
| 13 | \( 1 - 4.23iT - 13T^{2} \) |
| 17 | \( 1 - 1.85T + 17T^{2} \) |
| 23 | \( 1 - 4.55iT - 23T^{2} \) |
| 29 | \( 1 - 4.50iT - 29T^{2} \) |
| 31 | \( 1 + 2.95T + 31T^{2} \) |
| 37 | \( 1 + 2.49iT - 37T^{2} \) |
| 41 | \( 1 - 4.10iT - 41T^{2} \) |
| 43 | \( 1 - 0.970iT - 43T^{2} \) |
| 47 | \( 1 + 4.14iT - 47T^{2} \) |
| 53 | \( 1 - 14.2iT - 53T^{2} \) |
| 59 | \( 1 - 0.965T + 59T^{2} \) |
| 61 | \( 1 - 3.72T + 61T^{2} \) |
| 67 | \( 1 - 4.81T + 67T^{2} \) |
| 71 | \( 1 + 4.62T + 71T^{2} \) |
| 73 | \( 1 + 0.257T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 - 1.93iT - 83T^{2} \) |
| 89 | \( 1 + 2.54iT - 89T^{2} \) |
| 97 | \( 1 + 11.9iT - 97T^{2} \) |
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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
−8.241659313429011356422542281983, −7.50664233805282843644580647142, −6.98340477500502286012448383460, −6.16744260950453203532654151078, −5.50398272582277889402717438559, −4.49281398059194242680281734026, −3.82863565330849001627128630050, −3.28022309345798649558361441775, −1.63796521035566454261746028411, −0.77325376303069462853788389409,
0.50030967192418962958166145846, 2.03548682109222573512440051717, 2.70732022184948284041715501219, 3.82137469268550977400024115341, 4.85910298314446156595825108699, 5.21195751855628725705850993820, 6.09994819175421252949196675764, 6.79846242708918992262040311999, 7.59196143944283384732446524971, 8.266248253833743435863651827615