| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 3·7-s − 8-s + 9-s − 10-s − 11-s − 12-s − 3·14-s − 15-s + 16-s − 4·17-s − 18-s + 2·19-s + 20-s − 3·21-s + 22-s − 23-s + 24-s − 4·25-s − 27-s + 3·28-s − 9·29-s + 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.13·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.301·11-s − 0.288·12-s − 0.801·14-s − 0.258·15-s + 1/4·16-s − 0.970·17-s − 0.235·18-s + 0.458·19-s + 0.223·20-s − 0.654·21-s + 0.213·22-s − 0.208·23-s + 0.204·24-s − 4/5·25-s − 0.192·27-s + 0.566·28-s − 1.67·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11154 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11154 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + T + p T^{2} \) | 1.41.b |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−17.00271940762731, −16.09923709870537, −15.94644033960058, −14.99878492803243, −14.74128652438317, −13.84689394151190, −13.36530711948752, −12.70016693988934, −11.93319144570564, −11.38527189657613, −11.02536684592367, −10.47014118282388, −9.675623413102995, −9.276558315199704, −8.517765796631997, −7.718742275633715, −7.521347737301756, −6.524456536656092, −5.933276700002546, −5.330711032893972, −4.593165398905653, −3.873713980018231, −2.617920377385405, −1.941196199740374, −1.191647055698837, 0,
1.191647055698837, 1.941196199740374, 2.617920377385405, 3.873713980018231, 4.593165398905653, 5.330711032893972, 5.933276700002546, 6.524456536656092, 7.521347737301756, 7.718742275633715, 8.517765796631997, 9.276558315199704, 9.675623413102995, 10.47014118282388, 11.02536684592367, 11.38527189657613, 11.93319144570564, 12.70016693988934, 13.36530711948752, 13.84689394151190, 14.74128652438317, 14.99878492803243, 15.94644033960058, 16.09923709870537, 17.00271940762731
