| L(s) = 1 | + 2·5-s + 7-s + 2·11-s − 6·17-s − 4·19-s − 25-s − 6·29-s + 4·31-s + 2·35-s − 4·37-s + 6·41-s + 12·43-s + 2·47-s + 49-s − 2·53-s + 4·55-s + 10·59-s − 14·61-s − 8·67-s + 6·71-s + 2·77-s + 6·83-s − 12·85-s − 6·89-s − 8·95-s + 8·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.377·7-s + 0.603·11-s − 1.45·17-s − 0.917·19-s − 1/5·25-s − 1.11·29-s + 0.718·31-s + 0.338·35-s − 0.657·37-s + 0.937·41-s + 1.82·43-s + 0.291·47-s + 1/7·49-s − 0.274·53-s + 0.539·55-s + 1.30·59-s − 1.79·61-s − 0.977·67-s + 0.712·71-s + 0.227·77-s + 0.658·83-s − 1.30·85-s − 0.635·89-s − 0.820·95-s + 0.812·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 340704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 340704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.505691595\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.505691595\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−12.63162222424420, −12.14173386900390, −11.64009748321777, −11.04107302051208, −10.89106089685685, −10.33289852024026, −9.829275253783631, −9.259169421988043, −8.956542223159613, −8.691171979254810, −7.922109307718018, −7.500879465902323, −6.984445276613991, −6.358800449372370, −6.115089819552098, −5.675713372727571, −4.983181573693152, −4.502318490712039, −4.066234501926360, −3.583343572580843, −2.643903314595634, −2.290048586796681, −1.830231155045055, −1.224084131308136, −0.4036425578909417,
0.4036425578909417, 1.224084131308136, 1.830231155045055, 2.290048586796681, 2.643903314595634, 3.583343572580843, 4.066234501926360, 4.502318490712039, 4.983181573693152, 5.675713372727571, 6.115089819552098, 6.358800449372370, 6.984445276613991, 7.500879465902323, 7.922109307718018, 8.691171979254810, 8.956542223159613, 9.259169421988043, 9.829275253783631, 10.33289852024026, 10.89106089685685, 11.04107302051208, 11.64009748321777, 12.14173386900390, 12.63162222424420
