| L(s) = 1 | − 3-s − 3·5-s + 9-s − 3·11-s − 4·13-s + 3·15-s − 6·17-s + 19-s + 23-s + 4·25-s − 27-s − 2·29-s − 2·31-s + 3·33-s − 4·37-s + 4·39-s + 6·41-s + 3·43-s − 3·45-s + 9·47-s + 6·51-s + 6·53-s + 9·55-s − 57-s − 4·59-s − 3·61-s + 12·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s + 1/3·9-s − 0.904·11-s − 1.10·13-s + 0.774·15-s − 1.45·17-s + 0.229·19-s + 0.208·23-s + 4/5·25-s − 0.192·27-s − 0.371·29-s − 0.359·31-s + 0.522·33-s − 0.657·37-s + 0.640·39-s + 0.937·41-s + 0.457·43-s − 0.447·45-s + 1.31·47-s + 0.840·51-s + 0.824·53-s + 1.21·55-s − 0.132·57-s − 0.520·59-s − 0.384·61-s + 1.48·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 178752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 178752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 19 | \( 1 - T \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 3 T + p T^{2} \) | 1.43.ad |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 3 T + p T^{2} \) | 1.61.d |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 5 T + p T^{2} \) | 1.83.af |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−13.44631420923672, −13.19091846851223, −12.44536578810472, −12.23062975302982, −11.86148001413935, −11.20359594392995, −10.86420521161394, −10.58370010417324, −9.945054591811870, −9.246640684247395, −8.959858420412194, −8.283668137738305, −7.695244704079587, −7.439507700411455, −7.013494362496159, −6.465473106545645, −5.698509209536940, −5.288292781686014, −4.719080806117123, −4.213601026939702, −3.920492908237047, −2.996219652684435, −2.569553054807489, −1.909267782833338, −0.9326974465595497, 0, 0,
0.9326974465595497, 1.909267782833338, 2.569553054807489, 2.996219652684435, 3.920492908237047, 4.213601026939702, 4.719080806117123, 5.288292781686014, 5.698509209536940, 6.465473106545645, 7.013494362496159, 7.439507700411455, 7.695244704079587, 8.283668137738305, 8.959858420412194, 9.246640684247395, 9.945054591811870, 10.58370010417324, 10.86420521161394, 11.20359594392995, 11.86148001413935, 12.23062975302982, 12.44536578810472, 13.19091846851223, 13.44631420923672
