| L(s) = 1 | − 3-s − 2·5-s + 9-s − 2·11-s + 13-s + 2·15-s + 4·19-s + 8·23-s − 25-s − 27-s − 8·29-s − 4·31-s + 2·33-s + 4·37-s − 39-s − 6·41-s − 10·43-s − 2·45-s − 10·47-s + 4·55-s − 4·57-s + 4·59-s + 2·61-s − 2·65-s + 8·67-s − 8·69-s + 75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s − 0.603·11-s + 0.277·13-s + 0.516·15-s + 0.917·19-s + 1.66·23-s − 1/5·25-s − 0.192·27-s − 1.48·29-s − 0.718·31-s + 0.348·33-s + 0.657·37-s − 0.160·39-s − 0.937·41-s − 1.52·43-s − 0.298·45-s − 1.45·47-s + 0.539·55-s − 0.529·57-s + 0.520·59-s + 0.256·61-s − 0.248·65-s + 0.977·67-s − 0.963·69-s + 0.115·75-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 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−12.99900134990611, −12.75627748656640, −12.00690305780574, −11.69079405411097, −11.19324772080017, −11.03762613919907, −10.46287956885789, −9.729879409393801, −9.559498919188027, −8.905513702856924, −8.233613187793751, −7.989848822713569, −7.391833228446429, −6.951374929235332, −6.629624295999398, −5.796882007694422, −5.351439867398011, −5.014001330532359, −4.477037409697320, −3.643544338440765, −3.489794743175290, −2.848459493866724, −1.984519285232730, −1.401943438793734, −0.6285980543120645, 0,
0.6285980543120645, 1.401943438793734, 1.984519285232730, 2.848459493866724, 3.489794743175290, 3.643544338440765, 4.477037409697320, 5.014001330532359, 5.351439867398011, 5.796882007694422, 6.629624295999398, 6.951374929235332, 7.391833228446429, 7.989848822713569, 8.233613187793751, 8.905513702856924, 9.559498919188027, 9.729879409393801, 10.46287956885789, 11.03762613919907, 11.19324772080017, 11.69079405411097, 12.00690305780574, 12.75627748656640, 12.99900134990611
