| L(s) = 1 | − 3-s − 4·7-s + 9-s − 4·13-s + 8·19-s + 4·21-s + 4·23-s − 27-s + 6·29-s − 8·31-s + 4·37-s + 4·39-s + 6·41-s − 4·43-s − 4·47-s + 9·49-s + 12·53-s − 8·57-s + 6·61-s − 4·63-s − 12·67-s − 4·69-s − 16·71-s − 8·79-s + 81-s + 12·83-s − 6·87-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s − 1.10·13-s + 1.83·19-s + 0.872·21-s + 0.834·23-s − 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.657·37-s + 0.640·39-s + 0.937·41-s − 0.609·43-s − 0.583·47-s + 9/7·49-s + 1.64·53-s − 1.05·57-s + 0.768·61-s − 0.503·63-s − 1.46·67-s − 0.481·69-s − 1.89·71-s − 0.900·79-s + 1/9·81-s + 1.31·83-s − 0.643·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 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
−7.59677133623291837937243958322, −7.17070682462121059859449164270, −6.53423541397788765261975868051, −5.67215216768743677169052284106, −5.15521786178474173831229388408, −4.18178925982507798742366376859, −3.21062246766300642925020082451, −2.63960111586149698091714954769, −1.12503448299216464711851523823, 0,
1.12503448299216464711851523823, 2.63960111586149698091714954769, 3.21062246766300642925020082451, 4.18178925982507798742366376859, 5.15521786178474173831229388408, 5.67215216768743677169052284106, 6.53423541397788765261975868051, 7.17070682462121059859449164270, 7.59677133623291837937243958322
