| L(s) = 1 | + 3-s + 9-s + 11-s + 13-s + 17-s − 5·19-s + 4·23-s − 5·25-s + 27-s − 3·29-s + 33-s − 8·37-s + 39-s − 10·41-s + 4·43-s + 7·47-s + 51-s − 5·53-s − 5·57-s + 9·59-s − 3·61-s − 13·67-s + 4·69-s + 11·71-s − 14·73-s − 5·75-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 0.301·11-s + 0.277·13-s + 0.242·17-s − 1.14·19-s + 0.834·23-s − 25-s + 0.192·27-s − 0.557·29-s + 0.174·33-s − 1.31·37-s + 0.160·39-s − 1.56·41-s + 0.609·43-s + 1.02·47-s + 0.140·51-s − 0.686·53-s − 0.662·57-s + 1.17·59-s − 0.384·61-s − 1.58·67-s + 0.481·69-s + 1.30·71-s − 1.63·73-s − 0.577·75-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 244608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 244608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 + 5 T + p T^{2} \) | 1.53.f |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 + 3 T + p T^{2} \) | 1.61.d |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 - 11 T + p T^{2} \) | 1.71.al |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−13.19819628524819, −12.67071448988666, −12.14594085664697, −11.87435523623690, −11.14890724487157, −10.79067426415242, −10.31162709351536, −9.844742999214600, −9.312338289254927, −8.790618794910213, −8.580336949870833, −7.978538993846532, −7.455820386334843, −6.977292775892176, −6.567243056557744, −5.886228471829935, −5.535788636444304, −4.768927834893601, −4.371598602410923, −3.684684227546768, −3.402984415223617, −2.725731855672152, −1.956541490829656, −1.715915301519756, −0.8239462931440832, 0,
0.8239462931440832, 1.715915301519756, 1.956541490829656, 2.725731855672152, 3.402984415223617, 3.684684227546768, 4.371598602410923, 4.768927834893601, 5.535788636444304, 5.886228471829935, 6.567243056557744, 6.977292775892176, 7.455820386334843, 7.978538993846532, 8.580336949870833, 8.790618794910213, 9.312338289254927, 9.844742999214600, 10.31162709351536, 10.79067426415242, 11.14890724487157, 11.87435523623690, 12.14594085664697, 12.67071448988666, 13.19819628524819
