| L(s) = 1 | + 5-s + 7-s + 13-s + 7·17-s − 6·19-s − 2·23-s − 4·25-s − 4·29-s − 8·31-s + 35-s + 3·37-s − 6·41-s + 7·43-s + 3·47-s − 6·49-s + 6·53-s + 4·59-s + 10·61-s + 65-s − 14·67-s + 71-s + 4·73-s − 16·79-s − 4·83-s + 7·85-s + 2·89-s + 91-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s + 0.277·13-s + 1.69·17-s − 1.37·19-s − 0.417·23-s − 4/5·25-s − 0.742·29-s − 1.43·31-s + 0.169·35-s + 0.493·37-s − 0.937·41-s + 1.06·43-s + 0.437·47-s − 6/7·49-s + 0.824·53-s + 0.520·59-s + 1.28·61-s + 0.124·65-s − 1.71·67-s + 0.118·71-s + 0.468·73-s − 1.80·79-s − 0.439·83-s + 0.759·85-s + 0.211·89-s + 0.104·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14976 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
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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
−16.35520188361727, −15.90813927082118, −14.95244850076918, −14.74141897059784, −14.21040639701998, −13.53109563657570, −12.99359729167260, −12.47175058307134, −11.84412292493863, −11.25323328862359, −10.63134689699818, −10.09406940442275, −9.551724046304637, −8.882142923504258, −8.235091741708999, −7.694672191533670, −7.076139222157475, −6.239785701291641, −5.626411083078478, −5.292434543633167, −4.142836704362079, −3.800445870453489, −2.788391036271374, −1.953445163530711, −1.309429710216224, 0,
1.309429710216224, 1.953445163530711, 2.788391036271374, 3.800445870453489, 4.142836704362079, 5.292434543633167, 5.626411083078478, 6.239785701291641, 7.076139222157475, 7.694672191533670, 8.235091741708999, 8.882142923504258, 9.551724046304637, 10.09406940442275, 10.63134689699818, 11.25323328862359, 11.84412292493863, 12.47175058307134, 12.99359729167260, 13.53109563657570, 14.21040639701998, 14.74141897059784, 14.95244850076918, 15.90813927082118, 16.35520188361727
