| L(s) = 1 | − 3-s + 9-s − 4·11-s − 13-s − 6·17-s − 3·19-s + 6·23-s − 27-s + 4·33-s − 3·37-s + 39-s − 4·41-s + 6·47-s + 6·51-s − 6·53-s + 3·57-s − 2·59-s + 11·61-s − 7·67-s − 6·69-s + 4·71-s + 11·73-s − 15·79-s + 81-s + 6·83-s + 14·89-s − 97-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.20·11-s − 0.277·13-s − 1.45·17-s − 0.688·19-s + 1.25·23-s − 0.192·27-s + 0.696·33-s − 0.493·37-s + 0.160·39-s − 0.624·41-s + 0.875·47-s + 0.840·51-s − 0.824·53-s + 0.397·57-s − 0.260·59-s + 1.40·61-s − 0.855·67-s − 0.722·69-s + 0.474·71-s + 1.28·73-s − 1.68·79-s + 1/9·81-s + 0.658·83-s + 1.48·89-s − 0.101·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−14.69953274340330, −14.01275939478245, −13.39030273301593, −13.02362854552797, −12.71381488269134, −12.04045078294871, −11.48087347548884, −10.89290386033437, −10.68305043330379, −10.12987482837536, −9.470696876944605, −8.864739001828500, −8.453425329285656, −7.766715640044329, −7.172933802978743, −6.735024034241615, −6.182377901716852, −5.486721331550321, −4.935443004095359, −4.601184469713528, −3.832694940368682, −3.035681553964404, −2.377348990315556, −1.824156560152507, −0.7265329967598930, 0,
0.7265329967598930, 1.824156560152507, 2.377348990315556, 3.035681553964404, 3.832694940368682, 4.601184469713528, 4.935443004095359, 5.486721331550321, 6.182377901716852, 6.735024034241615, 7.172933802978743, 7.766715640044329, 8.453425329285656, 8.864739001828500, 9.470696876944605, 10.12987482837536, 10.68305043330379, 10.89290386033437, 11.48087347548884, 12.04045078294871, 12.71381488269134, 13.02362854552797, 13.39030273301593, 14.01275939478245, 14.69953274340330
