| L(s) = 1 | − 5-s + 4·7-s − 5·13-s − 4·17-s − 19-s − 4·23-s − 4·25-s + 10·29-s + 4·31-s − 4·35-s + 2·37-s − 8·41-s − 43-s − 4·47-s + 9·49-s − 53-s − 8·59-s − 14·61-s + 5·65-s + 12·67-s − 71-s − 2·73-s + 8·79-s − 83-s + 4·85-s + 12·89-s − 20·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s − 1.38·13-s − 0.970·17-s − 0.229·19-s − 0.834·23-s − 4/5·25-s + 1.85·29-s + 0.718·31-s − 0.676·35-s + 0.328·37-s − 1.24·41-s − 0.152·43-s − 0.583·47-s + 9/7·49-s − 0.137·53-s − 1.04·59-s − 1.79·61-s + 0.620·65-s + 1.46·67-s − 0.118·71-s − 0.234·73-s + 0.900·79-s − 0.109·83-s + 0.433·85-s + 1.27·89-s − 2.09·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 397 | \( 1 - T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + T + p T^{2} \) | 1.71.b |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + T + p T^{2} \) | 1.83.b |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 - 17 T + p T^{2} \) | 1.97.ar |
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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
−13.27942988037991, −12.48660268466932, −12.04272361344227, −11.90269047450117, −11.41271258834794, −10.84590288678135, −10.45631399310882, −9.940483499893094, −9.483339035190039, −8.829873584884827, −8.267436777362886, −8.077955270245115, −7.599039457009132, −7.099321430700498, −6.420814679129752, −6.134742048367640, −5.159175624210874, −4.893203931254631, −4.496060450779006, −4.089212872095628, −3.251183526663958, −2.628473342451952, −2.029353815670732, −1.662484297781412, −0.7286452633612579, 0,
0.7286452633612579, 1.662484297781412, 2.029353815670732, 2.628473342451952, 3.251183526663958, 4.089212872095628, 4.496060450779006, 4.893203931254631, 5.159175624210874, 6.134742048367640, 6.420814679129752, 7.099321430700498, 7.599039457009132, 8.077955270245115, 8.267436777362886, 8.829873584884827, 9.483339035190039, 9.940483499893094, 10.45631399310882, 10.84590288678135, 11.41271258834794, 11.90269047450117, 12.04272361344227, 12.48660268466932, 13.27942988037991
