| L(s) = 1 | + 3·3-s − 5-s + 4·7-s + 6·9-s − 2·11-s + 13-s − 3·15-s + 7·19-s + 12·21-s + 6·23-s + 25-s + 9·27-s − 3·29-s − 7·31-s − 6·33-s − 4·35-s − 2·37-s + 3·39-s + 8·41-s − 8·43-s − 6·45-s − 9·47-s + 9·49-s + 11·53-s + 2·55-s + 21·57-s − 5·59-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.447·5-s + 1.51·7-s + 2·9-s − 0.603·11-s + 0.277·13-s − 0.774·15-s + 1.60·19-s + 2.61·21-s + 1.25·23-s + 1/5·25-s + 1.73·27-s − 0.557·29-s − 1.25·31-s − 1.04·33-s − 0.676·35-s − 0.328·37-s + 0.480·39-s + 1.24·41-s − 1.21·43-s − 0.894·45-s − 1.31·47-s + 9/7·49-s + 1.51·53-s + 0.269·55-s + 2.78·57-s − 0.650·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.890879764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.890879764\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 - 11 T + p T^{2} \) | 1.53.al |
| 59 | \( 1 + 5 T + p T^{2} \) | 1.59.f |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + T + p T^{2} \) | 1.89.b |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−15.29754532908618, −14.90518942949716, −14.33965003141404, −14.08476234310231, −13.37039359378399, −13.01309687832236, −12.34258674377766, −11.46605886598766, −11.22795266834882, −10.52438295800943, −9.802631034408140, −9.196984045251210, −8.756075071483330, −8.190090984626307, −7.662885793948290, −7.453791292262354, −6.751500165563171, −5.435315833906167, −5.089617872381992, −4.370126144445224, −3.559415549445462, −3.199039965540034, −2.325925246861903, −1.700993199235205, −0.9428690515472716,
0.9428690515472716, 1.700993199235205, 2.325925246861903, 3.199039965540034, 3.559415549445462, 4.370126144445224, 5.089617872381992, 5.435315833906167, 6.751500165563171, 7.453791292262354, 7.662885793948290, 8.190090984626307, 8.756075071483330, 9.196984045251210, 9.802631034408140, 10.52438295800943, 11.22795266834882, 11.46605886598766, 12.34258674377766, 13.01309687832236, 13.37039359378399, 14.08476234310231, 14.33965003141404, 14.90518942949716, 15.29754532908618
