| L(s) = 1 | − 3-s − 5-s − 2·7-s + 9-s − 11-s + 13-s + 15-s + 6·19-s + 2·21-s − 4·25-s − 27-s − 6·29-s − 4·31-s + 33-s + 2·35-s + 10·37-s − 39-s + 5·41-s − 9·43-s − 45-s − 12·47-s − 3·49-s − 2·53-s + 55-s − 6·57-s − 3·59-s − 5·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.301·11-s + 0.277·13-s + 0.258·15-s + 1.37·19-s + 0.436·21-s − 4/5·25-s − 0.192·27-s − 1.11·29-s − 0.718·31-s + 0.174·33-s + 0.338·35-s + 1.64·37-s − 0.160·39-s + 0.780·41-s − 1.37·43-s − 0.149·45-s − 1.75·47-s − 3/7·49-s − 0.274·53-s + 0.134·55-s − 0.794·57-s − 0.390·59-s − 0.640·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 330096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 330096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4286849285\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4286849285\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| 23 | \( 1 \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 + 9 T + p T^{2} \) | 1.43.j |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 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.ag |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−12.51132439418247, −12.21991950666635, −11.50613239171170, −11.29581655276629, −11.03758766389607, −10.23037928371117, −9.870699590985384, −9.408676741517743, −9.221630840600521, −8.318983026466926, −7.863649857933346, −7.590278460293832, −7.019814100344611, −6.496246648400303, −6.011386779908250, −5.662926025404073, −4.991235996915642, −4.690097835850014, −3.829746704047281, −3.554589642177086, −3.072965120339718, −2.341039357500698, −1.638159249717619, −1.023542889025672, −0.2000155898269775,
0.2000155898269775, 1.023542889025672, 1.638159249717619, 2.341039357500698, 3.072965120339718, 3.554589642177086, 3.829746704047281, 4.690097835850014, 4.991235996915642, 5.662926025404073, 6.011386779908250, 6.496246648400303, 7.019814100344611, 7.590278460293832, 7.863649857933346, 8.318983026466926, 9.221630840600521, 9.408676741517743, 9.870699590985384, 10.23037928371117, 11.03758766389607, 11.29581655276629, 11.50613239171170, 12.21991950666635, 12.51132439418247
