| L(s) = 1 | − 3·5-s − 7-s + 6·11-s + 4·17-s − 3·19-s + 4·25-s − 2·29-s − 31-s + 3·35-s + 2·37-s + 41-s − 6·43-s + 4·47-s − 6·49-s − 12·53-s − 18·55-s − 7·59-s − 4·67-s + 15·71-s + 2·73-s − 6·77-s − 2·79-s − 6·83-s − 12·85-s + 14·89-s + 9·95-s − 7·97-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.377·7-s + 1.80·11-s + 0.970·17-s − 0.688·19-s + 4/5·25-s − 0.371·29-s − 0.179·31-s + 0.507·35-s + 0.328·37-s + 0.156·41-s − 0.914·43-s + 0.583·47-s − 6/7·49-s − 1.64·53-s − 2.42·55-s − 0.911·59-s − 0.488·67-s + 1.78·71-s + 0.234·73-s − 0.683·77-s − 0.225·79-s − 0.658·83-s − 1.30·85-s + 1.48·89-s + 0.923·95-s − 0.710·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 31 | \( 1 + T \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - T + p T^{2} \) | 1.41.ab |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 7 T + p T^{2} \) | 1.59.h |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−16.06240095165554, −15.57906867017557, −14.87831189277207, −14.61817742572496, −14.04575818679469, −13.30656212354079, −12.51039515259832, −12.23678481148690, −11.71817803424993, −11.15960751611443, −10.70612929468326, −9.674581244441436, −9.453112313422160, −8.658201499092506, −8.094265696044845, −7.582141726819922, −6.827412044250231, −6.428323047051401, −5.693312217865599, −4.705452538597478, −4.170215226501404, −3.550978069865081, −3.149504330590641, −1.867508825077614, −1.025110783592785, 0,
1.025110783592785, 1.867508825077614, 3.149504330590641, 3.550978069865081, 4.170215226501404, 4.705452538597478, 5.693312217865599, 6.428323047051401, 6.827412044250231, 7.582141726819922, 8.094265696044845, 8.658201499092506, 9.453112313422160, 9.674581244441436, 10.70612929468326, 11.15960751611443, 11.71817803424993, 12.23678481148690, 12.51039515259832, 13.30656212354079, 14.04575818679469, 14.61817742572496, 14.87831189277207, 15.57906867017557, 16.06240095165554
