| L(s) = 1 | + 5-s − 4·7-s + 13-s − 2·17-s − 4·19-s + 8·23-s + 25-s − 6·29-s − 10·31-s − 4·35-s − 8·37-s − 10·43-s + 8·47-s + 9·49-s − 8·53-s − 4·61-s + 65-s + 8·67-s − 8·71-s − 2·73-s + 4·79-s − 2·83-s − 2·85-s + 10·89-s − 4·91-s − 4·95-s + 12·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.51·7-s + 0.277·13-s − 0.485·17-s − 0.917·19-s + 1.66·23-s + 1/5·25-s − 1.11·29-s − 1.79·31-s − 0.676·35-s − 1.31·37-s − 1.52·43-s + 1.16·47-s + 9/7·49-s − 1.09·53-s − 0.512·61-s + 0.124·65-s + 0.977·67-s − 0.949·71-s − 0.234·73-s + 0.450·79-s − 0.219·83-s − 0.216·85-s + 1.05·89-s − 0.419·91-s − 0.410·95-s + 1.21·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 283140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−12.88959888782650, −12.77322859532356, −12.20559675442280, −11.53778423698760, −10.97574176896079, −10.72689244292060, −10.22069650014749, −9.694331003690227, −9.238982442227176, −8.815937664388486, −8.693481604272021, −7.700362621776147, −7.234806537812367, −6.815790376872235, −6.421792364718548, −5.959593514121512, −5.388215478408261, −4.972600250668387, −4.266736562996541, −3.526838743629294, −3.397343112411363, −2.736070966958344, −1.999186189430442, −1.619457215769039, −0.6001701326775535, 0,
0.6001701326775535, 1.619457215769039, 1.999186189430442, 2.736070966958344, 3.397343112411363, 3.526838743629294, 4.266736562996541, 4.972600250668387, 5.388215478408261, 5.959593514121512, 6.421792364718548, 6.815790376872235, 7.234806537812367, 7.700362621776147, 8.693481604272021, 8.815937664388486, 9.238982442227176, 9.694331003690227, 10.22069650014749, 10.72689244292060, 10.97574176896079, 11.53778423698760, 12.20559675442280, 12.77322859532356, 12.88959888782650
