| L(s) = 1 | − 4·7-s + 5·11-s − 13-s + 3·17-s + 6·19-s + 7·23-s − 7·29-s − 5·31-s + 6·37-s + 8·41-s − 9·43-s + 47-s + 9·49-s + 6·53-s + 2·61-s + 12·67-s − 6·71-s − 2·73-s − 20·77-s − 11·79-s + 14·83-s + 6·89-s + 4·91-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 1.50·11-s − 0.277·13-s + 0.727·17-s + 1.37·19-s + 1.45·23-s − 1.29·29-s − 0.898·31-s + 0.986·37-s + 1.24·41-s − 1.37·43-s + 0.145·47-s + 9/7·49-s + 0.824·53-s + 0.256·61-s + 1.46·67-s − 0.712·71-s − 0.234·73-s − 2.27·77-s − 1.23·79-s + 1.53·83-s + 0.635·89-s + 0.419·91-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.764152266\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.764152266\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 7 T + p T^{2} \) | 1.23.ah |
| 29 | \( 1 + 7 T + p T^{2} \) | 1.29.h |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 + 9 T + p T^{2} \) | 1.43.j |
| 47 | \( 1 - T + p T^{2} \) | 1.47.ab |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−13.11574317158421, −12.78374213898723, −12.33802288205823, −11.71104864368415, −11.44539341249945, −10.93883525175591, −10.13971027942503, −9.813384935858361, −9.314396872510843, −9.164531500847347, −8.604913955726185, −7.692313268660068, −7.287511000223592, −6.986455001271752, −6.350921505530867, −5.935291457263937, −5.402965060613746, −4.857517783562722, −4.006892972755632, −3.594169106937117, −3.232224878480634, −2.647850071424156, −1.794600382405104, −1.039586872488653, −0.5655069438280605,
0.5655069438280605, 1.039586872488653, 1.794600382405104, 2.647850071424156, 3.232224878480634, 3.594169106937117, 4.006892972755632, 4.857517783562722, 5.402965060613746, 5.935291457263937, 6.350921505530867, 6.986455001271752, 7.287511000223592, 7.692313268660068, 8.604913955726185, 9.164531500847347, 9.314396872510843, 9.813384935858361, 10.13971027942503, 10.93883525175591, 11.44539341249945, 11.71104864368415, 12.33802288205823, 12.78374213898723, 13.11574317158421
