| L(s) = 1 | − 5-s − 13-s − 4·17-s − 4·19-s + 4·23-s + 25-s + 4·29-s + 8·31-s − 6·37-s + 10·41-s + 12·43-s − 2·47-s − 7·49-s − 12·53-s − 2·59-s + 14·61-s + 65-s + 4·67-s − 6·71-s − 10·73-s + 8·79-s − 6·83-s + 4·85-s + 2·89-s + 4·95-s + 10·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.277·13-s − 0.970·17-s − 0.917·19-s + 0.834·23-s + 1/5·25-s + 0.742·29-s + 1.43·31-s − 0.986·37-s + 1.56·41-s + 1.82·43-s − 0.291·47-s − 49-s − 1.64·53-s − 0.260·59-s + 1.79·61-s + 0.124·65-s + 0.488·67-s − 0.712·71-s − 1.17·73-s + 0.900·79-s − 0.658·83-s + 0.433·85-s + 0.211·89-s + 0.410·95-s + 1.01·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 283140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−12.95612714936509, −12.52991278124256, −12.11344274574571, −11.52025687058796, −11.06916361178562, −10.86245559321900, −10.16939949743268, −9.857542567341291, −9.123887338376683, −8.805977175027860, −8.421890126705719, −7.775098770208028, −7.453470087015547, −6.821350453128880, −6.340056494446624, −6.108379943628913, −5.213371193021424, −4.740502010212297, −4.413546367698255, −3.869604124685707, −3.196439544528286, −2.593581189058565, −2.254422976281986, −1.364827678982380, −0.7340326807575133, 0,
0.7340326807575133, 1.364827678982380, 2.254422976281986, 2.593581189058565, 3.196439544528286, 3.869604124685707, 4.413546367698255, 4.740502010212297, 5.213371193021424, 6.108379943628913, 6.340056494446624, 6.821350453128880, 7.453470087015547, 7.775098770208028, 8.421890126705719, 8.805977175027860, 9.123887338376683, 9.857542567341291, 10.16939949743268, 10.86245559321900, 11.06916361178562, 11.52025687058796, 12.11344274574571, 12.52991278124256, 12.95612714936509
