| L(s) = 1 | − 5-s − 7-s − 3·9-s − 4·11-s + 2·13-s − 2·17-s + 19-s + 25-s + 10·29-s + 8·31-s + 35-s − 6·37-s + 10·41-s − 8·43-s + 3·45-s + 49-s + 10·53-s + 4·55-s − 4·59-s + 2·61-s + 3·63-s − 2·65-s + 12·67-s − 4·71-s − 2·73-s + 4·77-s − 12·79-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s − 9-s − 1.20·11-s + 0.554·13-s − 0.485·17-s + 0.229·19-s + 1/5·25-s + 1.85·29-s + 1.43·31-s + 0.169·35-s − 0.986·37-s + 1.56·41-s − 1.21·43-s + 0.447·45-s + 1/7·49-s + 1.37·53-s + 0.539·55-s − 0.520·59-s + 0.256·61-s + 0.377·63-s − 0.248·65-s + 1.46·67-s − 0.474·71-s − 0.234·73-s + 0.455·77-s − 1.35·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 19 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−16.90718671010724, −15.99367651499095, −15.74741583299089, −15.40138218943991, −14.40773239827050, −14.03425946078190, −13.36465691426381, −12.90453655709314, −12.12887363598725, −11.64661709078857, −11.08952524895211, −10.30556514241439, −10.07805991587164, −8.958980361113875, −8.512483547058005, −8.076473442803076, −7.292014379059171, −6.533457476263253, −5.974833162773394, −5.193982685300184, −4.597149460386197, −3.683282136436096, −2.864998603185397, −2.471483338622767, −0.9997946324539813, 0,
0.9997946324539813, 2.471483338622767, 2.864998603185397, 3.683282136436096, 4.597149460386197, 5.193982685300184, 5.974833162773394, 6.533457476263253, 7.292014379059171, 8.076473442803076, 8.512483547058005, 8.958980361113875, 10.07805991587164, 10.30556514241439, 11.08952524895211, 11.64661709078857, 12.12887363598725, 12.90453655709314, 13.36465691426381, 14.03425946078190, 14.40773239827050, 15.40138218943991, 15.74741583299089, 15.99367651499095, 16.90718671010724
