| L(s) = 1 | + 2·5-s − 4·7-s − 6·11-s + 13-s + 2·17-s − 6·19-s + 4·23-s − 25-s + 10·29-s + 8·31-s − 8·35-s + 6·37-s + 6·41-s − 10·43-s + 9·49-s − 6·53-s − 12·55-s + 10·59-s − 2·61-s + 2·65-s + 2·67-s + 4·71-s − 2·73-s + 24·77-s − 8·79-s + 14·83-s + 4·85-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.51·7-s − 1.80·11-s + 0.277·13-s + 0.485·17-s − 1.37·19-s + 0.834·23-s − 1/5·25-s + 1.85·29-s + 1.43·31-s − 1.35·35-s + 0.986·37-s + 0.937·41-s − 1.52·43-s + 9/7·49-s − 0.824·53-s − 1.61·55-s + 1.30·59-s − 0.256·61-s + 0.248·65-s + 0.244·67-s + 0.474·71-s − 0.234·73-s + 2.73·77-s − 0.900·79-s + 1.53·83-s + 0.433·85-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 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−16.20651074223571, −15.84931502494129, −15.37172607549108, −14.71486401834401, −13.89812300066647, −13.43825166920582, −12.97030585415408, −12.73326717524251, −11.99172241223742, −11.10100747588476, −10.42627669925170, −10.05398319295822, −9.765756804021163, −8.894703419008846, −8.287976414496193, −7.740056327876340, −6.711111863139085, −6.403388635842873, −5.837164769448394, −5.099492319599255, −4.453945723336481, −3.410313612767244, −2.681123994305399, −2.409841072109003, −1.034804605517043, 0,
1.034804605517043, 2.409841072109003, 2.681123994305399, 3.410313612767244, 4.453945723336481, 5.099492319599255, 5.837164769448394, 6.403388635842873, 6.711111863139085, 7.740056327876340, 8.287976414496193, 8.894703419008846, 9.765756804021163, 10.05398319295822, 10.42627669925170, 11.10100747588476, 11.99172241223742, 12.73326717524251, 12.97030585415408, 13.43825166920582, 13.89812300066647, 14.71486401834401, 15.37172607549108, 15.84931502494129, 16.20651074223571
