| L(s) = 1 | + 3-s − 4·5-s + 3·7-s + 9-s − 5·11-s + 3·13-s − 4·15-s − 3·17-s − 7·19-s + 3·21-s − 9·23-s + 11·25-s + 27-s + 2·31-s − 5·33-s − 12·35-s + 3·39-s + 6·41-s + 4·43-s − 4·45-s − 10·47-s + 2·49-s − 3·51-s − 3·53-s + 20·55-s − 7·57-s − 4·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.78·5-s + 1.13·7-s + 1/3·9-s − 1.50·11-s + 0.832·13-s − 1.03·15-s − 0.727·17-s − 1.60·19-s + 0.654·21-s − 1.87·23-s + 11/5·25-s + 0.192·27-s + 0.359·31-s − 0.870·33-s − 2.02·35-s + 0.480·39-s + 0.937·41-s + 0.609·43-s − 0.596·45-s − 1.45·47-s + 2/7·49-s − 0.420·51-s − 0.412·53-s + 2.69·55-s − 0.927·57-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 262848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 262848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 37 | \( 1 \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 6 T + p T^{2} \) | 1.67.g |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 13 T + p T^{2} \) | 1.73.an |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + 5 T + p T^{2} \) | 1.83.f |
| 89 | \( 1 + 11 T + p T^{2} \) | 1.89.l |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−13.15589963636937, −12.80391488253648, −12.33592069070793, −11.91444873929450, −11.24187341307260, −11.04071601234196, −10.69586351618569, −10.20222331808806, −9.521710220898103, −8.721479213668631, −8.470477411176336, −8.074272794688674, −7.871909938279556, −7.461212337325038, −6.766328654319077, −6.217663563438605, −5.647474323767021, −4.804256291520023, −4.516401358651552, −4.148618656670013, −3.661241063303117, −2.986799554930917, −2.396331888949074, −1.898353567745416, −1.144401625014415, 0, 0,
1.144401625014415, 1.898353567745416, 2.396331888949074, 2.986799554930917, 3.661241063303117, 4.148618656670013, 4.516401358651552, 4.804256291520023, 5.647474323767021, 6.217663563438605, 6.766328654319077, 7.461212337325038, 7.871909938279556, 8.074272794688674, 8.470477411176336, 8.721479213668631, 9.521710220898103, 10.20222331808806, 10.69586351618569, 11.04071601234196, 11.24187341307260, 11.91444873929450, 12.33592069070793, 12.80391488253648, 13.15589963636937
