| L(s) = 1 | − 2·3-s − 2·7-s + 9-s − 2·13-s − 6·17-s + 4·19-s + 4·21-s − 6·23-s + 4·27-s + 4·31-s + 2·37-s + 4·39-s − 6·41-s − 10·43-s − 6·47-s − 3·49-s + 12·51-s + 6·53-s − 8·57-s + 12·59-s − 2·61-s − 2·63-s − 2·67-s + 12·69-s − 12·71-s + 2·73-s − 8·79-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.755·7-s + 1/3·9-s − 0.554·13-s − 1.45·17-s + 0.917·19-s + 0.872·21-s − 1.25·23-s + 0.769·27-s + 0.718·31-s + 0.328·37-s + 0.640·39-s − 0.937·41-s − 1.52·43-s − 0.875·47-s − 3/7·49-s + 1.68·51-s + 0.824·53-s − 1.05·57-s + 1.56·59-s − 0.256·61-s − 0.251·63-s − 0.244·67-s + 1.44·69-s − 1.42·71-s + 0.234·73-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 29 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−14.55727628892553, −13.70839139184115, −13.38708726349055, −13.01305644271432, −12.19967600119508, −11.99510532633603, −11.47665526767087, −11.15679548166898, −10.36420054955300, −9.972176126997816, −9.730348774359156, −8.866735099495280, −8.459131243959761, −7.822139202497788, −7.035608748541179, −6.707108740061615, −6.268185388572471, −5.700579712107886, −5.151909112237315, −4.647117758311297, −4.065417719124932, −3.295407495333788, −2.700909639602699, −1.970813116558718, −1.122723999434891, 0, 0,
1.122723999434891, 1.970813116558718, 2.700909639602699, 3.295407495333788, 4.065417719124932, 4.647117758311297, 5.151909112237315, 5.700579712107886, 6.268185388572471, 6.707108740061615, 7.035608748541179, 7.822139202497788, 8.459131243959761, 8.866735099495280, 9.730348774359156, 9.972176126997816, 10.36420054955300, 11.15679548166898, 11.47665526767087, 11.99510532633603, 12.19967600119508, 13.01305644271432, 13.38708726349055, 13.70839139184115, 14.55727628892553
