| L(s) = 1 | − 3-s − 3·5-s − 2·7-s − 2·9-s − 11-s + 3·15-s + 6·17-s + 8·19-s + 2·21-s − 3·23-s + 4·25-s + 5·27-s − 5·31-s + 33-s + 6·35-s − 37-s + 10·43-s + 6·45-s − 3·49-s − 6·51-s + 6·53-s + 3·55-s − 8·57-s + 3·59-s + 4·61-s + 4·63-s − 67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s − 0.755·7-s − 2/3·9-s − 0.301·11-s + 0.774·15-s + 1.45·17-s + 1.83·19-s + 0.436·21-s − 0.625·23-s + 4/5·25-s + 0.962·27-s − 0.898·31-s + 0.174·33-s + 1.01·35-s − 0.164·37-s + 1.52·43-s + 0.894·45-s − 3/7·49-s − 0.840·51-s + 0.824·53-s + 0.404·55-s − 1.05·57-s + 0.390·59-s + 0.512·61-s + 0.503·63-s − 0.122·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 118976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 118976 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + T + p T^{2} \) | 1.67.b |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−13.89912677095027, −13.24000591938318, −12.70042438227955, −12.11843102113793, −11.95145157681267, −11.55013606489562, −11.02905237212396, −10.41888646319311, −10.06829233686186, −9.311024326946556, −9.079855258205871, −8.105568965780942, −7.965011317860156, −7.337383966797024, −7.038131281557294, −6.184150366714499, −5.704656225807503, −5.339224505202104, −4.759890026343445, −3.871689830492868, −3.574423646588467, −3.051264243383375, −2.484762796927291, −1.285313802664320, −0.6679287620690538, 0,
0.6679287620690538, 1.285313802664320, 2.484762796927291, 3.051264243383375, 3.574423646588467, 3.871689830492868, 4.759890026343445, 5.339224505202104, 5.704656225807503, 6.184150366714499, 7.038131281557294, 7.337383966797024, 7.965011317860156, 8.105568965780942, 9.079855258205871, 9.311024326946556, 10.06829233686186, 10.41888646319311, 11.02905237212396, 11.55013606489562, 11.95145157681267, 12.11843102113793, 12.70042438227955, 13.24000591938318, 13.89912677095027
