| L(s) = 1 | + 2-s + 4-s + 8-s − 4·11-s + 16-s + 2·19-s − 4·22-s + 4·23-s − 5·25-s + 6·29-s + 4·31-s + 32-s + 4·37-s + 2·38-s + 41-s − 4·44-s + 4·46-s − 12·47-s − 7·49-s − 5·50-s − 4·53-s + 6·58-s − 4·59-s − 4·61-s + 4·62-s + 64-s − 10·67-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 1.20·11-s + 1/4·16-s + 0.458·19-s − 0.852·22-s + 0.834·23-s − 25-s + 1.11·29-s + 0.718·31-s + 0.176·32-s + 0.657·37-s + 0.324·38-s + 0.156·41-s − 0.603·44-s + 0.589·46-s − 1.75·47-s − 49-s − 0.707·50-s − 0.549·53-s + 0.787·58-s − 0.520·59-s − 0.512·61-s + 0.508·62-s + 1/8·64-s − 1.22·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 213282 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 213282 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.311003169\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.311003169\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 17 | \( 1 \) | |
| 41 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 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.e |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−13.11898927971744, −12.65414266128499, −11.96715430326712, −11.83281459553823, −11.06931328804581, −10.82653552551341, −10.29074177511955, −9.662360242661951, −9.496848664989063, −8.622277057428615, −8.047427667977907, −7.848202469776113, −7.304366497525893, −6.535707092830068, −6.344923538048387, −5.684514912309648, −5.062654880385350, −4.826802412928869, −4.300512625807392, −3.461492480229877, −3.096815368149433, −2.591357010301478, −1.957917939043412, −1.247846169795431, −0.4495220617395049,
0.4495220617395049, 1.247846169795431, 1.957917939043412, 2.591357010301478, 3.096815368149433, 3.461492480229877, 4.300512625807392, 4.826802412928869, 5.062654880385350, 5.684514912309648, 6.344923538048387, 6.535707092830068, 7.304366497525893, 7.848202469776113, 8.047427667977907, 8.622277057428615, 9.496848664989063, 9.662360242661951, 10.29074177511955, 10.82653552551341, 11.06931328804581, 11.83281459553823, 11.96715430326712, 12.65414266128499, 13.11898927971744
