| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s − 5·11-s + 2·13-s + 15-s − 4·17-s − 19-s + 21-s + 5·23-s + 25-s − 27-s + 4·29-s + 31-s + 5·33-s + 35-s + 12·37-s − 2·39-s + 4·41-s − 11·43-s − 45-s + 10·47-s − 6·49-s + 4·51-s + 9·53-s + 5·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.50·11-s + 0.554·13-s + 0.258·15-s − 0.970·17-s − 0.229·19-s + 0.218·21-s + 1.04·23-s + 1/5·25-s − 0.192·27-s + 0.742·29-s + 0.179·31-s + 0.870·33-s + 0.169·35-s + 1.97·37-s − 0.320·39-s + 0.624·41-s − 1.67·43-s − 0.149·45-s + 1.45·47-s − 6/7·49-s + 0.560·51-s + 1.23·53-s + 0.674·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 31 | \( 1 - T \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 37 | \( 1 - 12 T + p T^{2} \) | 1.37.am |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 6 T + p T^{2} \) | 1.67.g |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 + 13 T + p T^{2} \) | 1.73.n |
| 79 | \( 1 + 13 T + p T^{2} \) | 1.79.n |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−7.40340133476372347634970755773, −6.91672393684671679611695104289, −6.10909844786154228141633026431, −5.48950169569923591166682060514, −4.67503384551942401815896801926, −4.13192679984659623895211858994, −3.01909549984983506404132637626, −2.42394062504032981721463063013, −1.02093953506516624660097004270, 0,
1.02093953506516624660097004270, 2.42394062504032981721463063013, 3.01909549984983506404132637626, 4.13192679984659623895211858994, 4.67503384551942401815896801926, 5.48950169569923591166682060514, 6.10909844786154228141633026431, 6.91672393684671679611695104289, 7.40340133476372347634970755773
