| L(s) = 1 | + 2·5-s − 4·11-s + 2·13-s + 2·17-s − 4·19-s − 25-s + 6·29-s + 8·31-s + 6·37-s + 6·41-s + 4·43-s − 7·49-s + 2·53-s − 8·55-s + 4·59-s − 2·61-s + 4·65-s − 4·67-s − 8·71-s + 10·73-s + 8·79-s + 4·83-s + 4·85-s − 6·89-s − 8·95-s − 2·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.20·11-s + 0.554·13-s + 0.485·17-s − 0.917·19-s − 1/5·25-s + 1.11·29-s + 1.43·31-s + 0.986·37-s + 0.937·41-s + 0.609·43-s − 49-s + 0.274·53-s − 1.07·55-s + 0.520·59-s − 0.256·61-s + 0.496·65-s − 0.488·67-s − 0.949·71-s + 1.17·73-s + 0.900·79-s + 0.439·83-s + 0.433·85-s − 0.635·89-s − 0.820·95-s − 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + p T^{2} \) | 1.7.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.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−12.82858360343024, −12.61620170989639, −12.05463624237779, −11.49727782589716, −10.89061016001097, −10.67671837370493, −10.04758243245368, −9.849562740803928, −9.329717747399086, −8.676879290223575, −8.278219959995740, −7.880281036094166, −7.426596467499857, −6.638195070647820, −6.262406695694476, −5.934254023288147, −5.377862754965742, −4.854359922680246, −4.376931202754515, −3.808031874727769, −3.026618464772507, −2.541026523316580, −2.261172818469708, −1.371989622560568, −0.8881185736399002, 0,
0.8881185736399002, 1.371989622560568, 2.261172818469708, 2.541026523316580, 3.026618464772507, 3.808031874727769, 4.376931202754515, 4.854359922680246, 5.377862754965742, 5.934254023288147, 6.262406695694476, 6.638195070647820, 7.426596467499857, 7.880281036094166, 8.278219959995740, 8.676879290223575, 9.329717747399086, 9.849562740803928, 10.04758243245368, 10.67671837370493, 10.89061016001097, 11.49727782589716, 12.05463624237779, 12.61620170989639, 12.82858360343024
