| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s − 3·11-s − 12-s − 15-s + 16-s − 18-s − 5·19-s + 20-s + 3·22-s + 6·23-s + 24-s + 25-s − 27-s − 6·29-s + 30-s + 7·31-s − 32-s + 3·33-s + 36-s − 4·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.904·11-s − 0.288·12-s − 0.258·15-s + 1/4·16-s − 0.235·18-s − 1.14·19-s + 0.223·20-s + 0.639·22-s + 1.25·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.182·30-s + 1.25·31-s − 0.176·32-s + 0.522·33-s + 1/6·36-s − 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
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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
−13.05033334392464, −12.61083580031876, −12.06220750722210, −11.71023092618224, −10.98257246238105, −10.70222219904722, −10.47644336607912, −9.860797500249256, −9.427456679341262, −8.904205467223496, −8.445738854935667, −8.014062941055955, −7.311746856299670, −7.071143970568787, −6.414472093947133, −6.041637925923837, −5.495495233834082, −4.977004518951128, −4.540068904184194, −3.828081107420511, −3.033327806334227, −2.673501901173879, −1.901385589687557, −1.482117214951421, −0.6375212157332078, 0,
0.6375212157332078, 1.482117214951421, 1.901385589687557, 2.673501901173879, 3.033327806334227, 3.828081107420511, 4.540068904184194, 4.977004518951128, 5.495495233834082, 6.041637925923837, 6.414472093947133, 7.071143970568787, 7.311746856299670, 8.014062941055955, 8.445738854935667, 8.904205467223496, 9.427456679341262, 9.860797500249256, 10.47644336607912, 10.70222219904722, 10.98257246238105, 11.71023092618224, 12.06220750722210, 12.61083580031876, 13.05033334392464
