| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s + 4·11-s − 13-s + 15-s + 19-s + 21-s + 2·23-s + 25-s − 27-s − 6·29-s − 5·31-s − 4·33-s + 35-s − 3·37-s + 39-s − 7·43-s − 45-s − 6·47-s − 6·49-s − 2·53-s − 4·55-s − 57-s − 14·59-s − 5·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.277·13-s + 0.258·15-s + 0.229·19-s + 0.218·21-s + 0.417·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 0.898·31-s − 0.696·33-s + 0.169·35-s − 0.493·37-s + 0.160·39-s − 1.06·43-s − 0.149·45-s − 0.875·47-s − 6/7·49-s − 0.274·53-s − 0.539·55-s − 0.132·57-s − 1.82·59-s − 0.640·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5218080373\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5218080373\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−14.24413497443879, −13.55708444226403, −13.06023580968800, −12.62811814622432, −11.99500074278150, −11.74687571141512, −11.16176657681038, −10.78302623623713, −10.06457079341247, −9.603112165600928, −9.065579431053246, −8.687216695147362, −7.850944170605295, −7.297607146338473, −6.981593492609857, −6.230971316643363, −5.945952542334142, −5.126385152640343, −4.628005746922272, −4.038976866292467, −3.364446625189622, −2.983438042846971, −1.695596325181579, −1.460992039205084, −0.2501326659440727,
0.2501326659440727, 1.460992039205084, 1.695596325181579, 2.983438042846971, 3.364446625189622, 4.038976866292467, 4.628005746922272, 5.126385152640343, 5.945952542334142, 6.230971316643363, 6.981593492609857, 7.297607146338473, 7.850944170605295, 8.687216695147362, 9.065579431053246, 9.603112165600928, 10.06457079341247, 10.78302623623713, 11.16176657681038, 11.74687571141512, 11.99500074278150, 12.62811814622432, 13.06023580968800, 13.55708444226403, 14.24413497443879
