| L(s) = 1 | − 5-s − 3·9-s − 2·11-s + 4·13-s + 6·17-s − 8·19-s + 23-s + 25-s − 10·29-s − 10·31-s − 8·37-s + 2·41-s + 3·45-s − 12·47-s + 4·53-s + 2·55-s + 14·59-s − 2·61-s − 4·65-s + 4·67-s + 8·71-s + 6·79-s + 9·81-s − 12·83-s − 6·85-s − 10·89-s + 8·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 9-s − 0.603·11-s + 1.10·13-s + 1.45·17-s − 1.83·19-s + 0.208·23-s + 1/5·25-s − 1.85·29-s − 1.79·31-s − 1.31·37-s + 0.312·41-s + 0.447·45-s − 1.75·47-s + 0.549·53-s + 0.269·55-s + 1.82·59-s − 0.256·61-s − 0.496·65-s + 0.488·67-s + 0.949·71-s + 0.675·79-s + 81-s − 1.31·83-s − 0.650·85-s − 1.05·89-s + 0.820·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−12.74026124677772, −12.50159582611397, −11.66299837136868, −11.32323970513519, −11.02929906635061, −10.64189999627554, −10.05094083865233, −9.638404982909467, −8.810461729517060, −8.693139438962015, −8.307226658778917, −7.676903200601633, −7.362770810915505, −6.742620455849524, −6.136360249181253, −5.773013802398839, −5.264775897080149, −4.963202354187655, −3.974431143846363, −3.625564163452590, −3.421070805890217, −2.598427342178778, −1.995815015922018, −1.519879341240405, −0.5606867502461132, 0,
0.5606867502461132, 1.519879341240405, 1.995815015922018, 2.598427342178778, 3.421070805890217, 3.625564163452590, 3.974431143846363, 4.963202354187655, 5.264775897080149, 5.773013802398839, 6.136360249181253, 6.742620455849524, 7.362770810915505, 7.676903200601633, 8.307226658778917, 8.693139438962015, 8.810461729517060, 9.638404982909467, 10.05094083865233, 10.64189999627554, 11.02929906635061, 11.32323970513519, 11.66299837136868, 12.50159582611397, 12.74026124677772
