| L(s) = 1 | − 2·5-s − 2·13-s + 6·17-s − 4·19-s − 8·23-s − 25-s + 2·29-s − 31-s − 10·37-s + 6·41-s − 8·43-s + 8·47-s − 7·49-s − 6·53-s − 12·59-s + 6·61-s + 4·65-s + 12·67-s − 8·71-s + 10·73-s − 8·79-s + 8·83-s − 12·85-s + 6·89-s + 8·95-s + 2·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.554·13-s + 1.45·17-s − 0.917·19-s − 1.66·23-s − 1/5·25-s + 0.371·29-s − 0.179·31-s − 1.64·37-s + 0.937·41-s − 1.21·43-s + 1.16·47-s − 49-s − 0.824·53-s − 1.56·59-s + 0.768·61-s + 0.496·65-s + 1.46·67-s − 0.949·71-s + 1.17·73-s − 0.900·79-s + 0.878·83-s − 1.30·85-s + 0.635·89-s + 0.820·95-s + 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9141121487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9141121487\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 \) | |
| 3 | \( 1 \) | |
| 31 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−15.73315037815487, −15.44243596329670, −14.61831633489857, −14.25068071234596, −13.80925050363028, −12.83908738652050, −12.42190895755218, −11.98229703306645, −11.52102426278714, −10.77666120103153, −10.12459263316262, −9.831771003336548, −8.923231059874582, −8.330475620832627, −7.727337954724499, −7.477972471626688, −6.516985878810531, −6.009904427587417, −5.186022363609855, −4.592322904339529, −3.755251247846160, −3.432023175606139, −2.377593892796587, −1.609981061966868, −0.3970655343119076,
0.3970655343119076, 1.609981061966868, 2.377593892796587, 3.432023175606139, 3.755251247846160, 4.592322904339529, 5.186022363609855, 6.009904427587417, 6.516985878810531, 7.477972471626688, 7.727337954724499, 8.330475620832627, 8.923231059874582, 9.831771003336548, 10.12459263316262, 10.77666120103153, 11.52102426278714, 11.98229703306645, 12.42190895755218, 12.83908738652050, 13.80925050363028, 14.25068071234596, 14.61831633489857, 15.44243596329670, 15.73315037815487
