| L(s) = 1 | + 5-s + 2·7-s + 2·11-s − 2·13-s + 2·17-s + 2·19-s − 2·23-s + 25-s + 6·29-s + 4·31-s + 2·35-s + 2·37-s + 10·41-s − 8·43-s − 2·47-s − 3·49-s − 6·53-s + 2·55-s − 2·59-s − 10·61-s − 2·65-s + 8·67-s + 8·71-s − 6·73-s + 4·77-s + 16·79-s + 12·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.755·7-s + 0.603·11-s − 0.554·13-s + 0.485·17-s + 0.458·19-s − 0.417·23-s + 1/5·25-s + 1.11·29-s + 0.718·31-s + 0.338·35-s + 0.328·37-s + 1.56·41-s − 1.21·43-s − 0.291·47-s − 3/7·49-s − 0.824·53-s + 0.269·55-s − 0.260·59-s − 1.28·61-s − 0.248·65-s + 0.977·67-s + 0.949·71-s − 0.702·73-s + 0.455·77-s + 1.80·79-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.620533817\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.620533817\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−8.003283416293369467106045226942, −7.58032014435231381189302135809, −6.54514557845037561239123661579, −6.10881509260436238779041595998, −5.04718712383865571252012090799, −4.69512401337153497182389132984, −3.65208714264992503032906694543, −2.73603985887109983766691810797, −1.81678134669354500784128367893, −0.905007767632968015563896492703,
0.905007767632968015563896492703, 1.81678134669354500784128367893, 2.73603985887109983766691810797, 3.65208714264992503032906694543, 4.69512401337153497182389132984, 5.04718712383865571252012090799, 6.10881509260436238779041595998, 6.54514557845037561239123661579, 7.58032014435231381189302135809, 8.003283416293369467106045226942
