| L(s) = 1 | − 3-s − 3·5-s + 4·7-s + 9-s − 13-s + 3·15-s + 17-s − 19-s − 4·21-s + 9·23-s + 4·25-s − 27-s + 6·29-s + 2·31-s − 12·35-s + 4·37-s + 39-s + 3·41-s − 7·43-s − 3·45-s − 6·47-s + 9·49-s − 51-s + 6·53-s + 57-s − 6·59-s + 8·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s + 1.51·7-s + 1/3·9-s − 0.277·13-s + 0.774·15-s + 0.242·17-s − 0.229·19-s − 0.872·21-s + 1.87·23-s + 4/5·25-s − 0.192·27-s + 1.11·29-s + 0.359·31-s − 2.02·35-s + 0.657·37-s + 0.160·39-s + 0.468·41-s − 1.06·43-s − 0.447·45-s − 0.875·47-s + 9/7·49-s − 0.140·51-s + 0.824·53-s + 0.132·57-s − 0.781·59-s + 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 394944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 394944 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−12.54987546549982, −12.14291056731399, −11.58584144849214, −11.36515762213871, −11.07645678161443, −10.65207989819258, −10.02482238716172, −9.622564347960882, −8.785136030961291, −8.539954821970916, −8.080257865923968, −7.674031323581306, −7.258939176462066, −6.761548187552285, −6.339027973112722, −5.509513772815292, −5.088993919329893, −4.689156991654245, −4.435095516347060, −3.753658532408727, −3.247030841083206, −2.585514791805842, −1.960393321322150, −1.054893534107099, −0.9238386970547174, 0,
0.9238386970547174, 1.054893534107099, 1.960393321322150, 2.585514791805842, 3.247030841083206, 3.753658532408727, 4.435095516347060, 4.689156991654245, 5.088993919329893, 5.509513772815292, 6.339027973112722, 6.761548187552285, 7.258939176462066, 7.674031323581306, 8.080257865923968, 8.539954821970916, 8.785136030961291, 9.622564347960882, 10.02482238716172, 10.65207989819258, 11.07645678161443, 11.36515762213871, 11.58584144849214, 12.14291056731399, 12.54987546549982
