| 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.ad |
| 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.b |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 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.d |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 3 T + p T^{2} \) | 1.61.d |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 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.f |
| 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
−13.34957605600209, −12.91628504592399, −12.35122852313343, −12.06702876181032, −11.52451689470311, −11.12063421806160, −10.73925329810828, −9.900665382159364, −9.618127952469407, −9.048424982271678, −8.570206010180050, −8.129671680757052, −7.751444590741335, −7.024269295754003, −6.861809860523160, −6.337269608362754, −5.456792355058474, −4.794242311797526, −4.448347948877143, −3.792211109162244, −3.645068504343535, −2.733481157719148, −2.274169376592308, −1.618752202779818, −0.6691623601684185, 0,
0.6691623601684185, 1.618752202779818, 2.274169376592308, 2.733481157719148, 3.645068504343535, 3.792211109162244, 4.448347948877143, 4.794242311797526, 5.456792355058474, 6.337269608362754, 6.861809860523160, 7.024269295754003, 7.751444590741335, 8.129671680757052, 8.570206010180050, 9.048424982271678, 9.618127952469407, 9.900665382159364, 10.73925329810828, 11.12063421806160, 11.52451689470311, 12.06702876181032, 12.35122852313343, 12.91628504592399, 13.34957605600209
