| L(s) = 1 | + 3·5-s − 3·7-s + 5·11-s + 6·13-s − 2·17-s + 4·19-s − 9·23-s + 4·25-s + 8·29-s − 10·31-s − 9·35-s + 37-s − 5·41-s + 4·43-s − 5·47-s + 2·49-s + 9·53-s + 15·55-s − 2·59-s + 4·61-s + 18·65-s − 14·67-s − 14·71-s + 73-s − 15·77-s + 2·79-s + 5·83-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 1.13·7-s + 1.50·11-s + 1.66·13-s − 0.485·17-s + 0.917·19-s − 1.87·23-s + 4/5·25-s + 1.48·29-s − 1.79·31-s − 1.52·35-s + 0.164·37-s − 0.780·41-s + 0.609·43-s − 0.729·47-s + 2/7·49-s + 1.23·53-s + 2.02·55-s − 0.260·59-s + 0.512·61-s + 2.23·65-s − 1.71·67-s − 1.66·71-s + 0.117·73-s − 1.70·77-s + 0.225·79-s + 0.548·83-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
| \(L(1)\) |
\(\approx\) |
\(3.634981251\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.634981251\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 5 T + p T^{2} \) | 1.47.f |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 - 5 T + p T^{2} \) | 1.83.af |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−13.19738422664825, −12.49901774423181, −11.96415638103585, −11.65839359448463, −11.07055572519302, −10.37013914379366, −10.16914257773816, −9.638014914898897, −9.213782148330172, −8.836455741048142, −8.500209987126620, −7.666239676826456, −7.062314822404227, −6.480714450633932, −6.252375802684230, −5.888726420037769, −5.489794790540391, −4.582069432884038, −4.031368026579984, −3.507907210829493, −3.173507618771948, −2.307342918319738, −1.703519481069798, −1.326837153179750, −0.5272087188706848,
0.5272087188706848, 1.326837153179750, 1.703519481069798, 2.307342918319738, 3.173507618771948, 3.507907210829493, 4.031368026579984, 4.582069432884038, 5.489794790540391, 5.888726420037769, 6.252375802684230, 6.480714450633932, 7.062314822404227, 7.666239676826456, 8.500209987126620, 8.836455741048142, 9.213782148330172, 9.638014914898897, 10.16914257773816, 10.37013914379366, 11.07055572519302, 11.65839359448463, 11.96415638103585, 12.49901774423181, 13.19738422664825
