| L(s) = 1 | + 3-s − 2·7-s + 9-s − 2·11-s − 2·17-s + 6·19-s − 2·21-s − 5·25-s + 27-s − 2·29-s + 2·31-s − 2·33-s + 4·37-s + 4·41-s − 4·43-s + 10·47-s − 3·49-s − 2·51-s − 10·53-s + 6·57-s + 6·59-s − 2·61-s − 2·63-s − 2·67-s + 6·71-s − 16·73-s − 5·75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.603·11-s − 0.485·17-s + 1.37·19-s − 0.436·21-s − 25-s + 0.192·27-s − 0.371·29-s + 0.359·31-s − 0.348·33-s + 0.657·37-s + 0.624·41-s − 0.609·43-s + 1.45·47-s − 3/7·49-s − 0.280·51-s − 1.37·53-s + 0.794·57-s + 0.781·59-s − 0.256·61-s − 0.251·63-s − 0.244·67-s + 0.712·71-s − 1.87·73-s − 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−7.56526351377024367495497491264, −6.90897456003563183018485863780, −6.07627655152921536874817650319, −5.46985738168407771189340811628, −4.56136063422693138051514835247, −3.79210471005439639529884270948, −3.02835896382273170418598635817, −2.43147204997704741306220331734, −1.31479236282188432012116140194, 0,
1.31479236282188432012116140194, 2.43147204997704741306220331734, 3.02835896382273170418598635817, 3.79210471005439639529884270948, 4.56136063422693138051514835247, 5.46985738168407771189340811628, 6.07627655152921536874817650319, 6.90897456003563183018485863780, 7.56526351377024367495497491264
