| L(s) = 1 | − 2·3-s − 4·7-s + 9-s − 11-s − 6·13-s + 2·17-s − 4·19-s + 8·21-s − 6·23-s + 4·27-s − 2·29-s − 8·31-s + 2·33-s − 8·37-s + 12·39-s + 6·41-s − 12·43-s − 10·47-s + 9·49-s − 4·51-s + 8·57-s + 4·59-s − 10·61-s − 4·63-s − 2·67-s + 12·69-s + 8·71-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.51·7-s + 1/3·9-s − 0.301·11-s − 1.66·13-s + 0.485·17-s − 0.917·19-s + 1.74·21-s − 1.25·23-s + 0.769·27-s − 0.371·29-s − 1.43·31-s + 0.348·33-s − 1.31·37-s + 1.92·39-s + 0.937·41-s − 1.82·43-s − 1.45·47-s + 9/7·49-s − 0.560·51-s + 1.05·57-s + 0.520·59-s − 1.28·61-s − 0.503·63-s − 0.244·67-s + 1.44·69-s + 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 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 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
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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
−7.39670640039587408191046820043, −6.76524121744274112910508903739, −6.17312501935864596969907723956, −5.47088967195342583565929028119, −4.85974732655216967430322002334, −3.80606683850703192972973716479, −2.96142857643921751826612429350, −1.93482862156308380196244786119, 0, 0,
1.93482862156308380196244786119, 2.96142857643921751826612429350, 3.80606683850703192972973716479, 4.85974732655216967430322002334, 5.47088967195342583565929028119, 6.17312501935864596969907723956, 6.76524121744274112910508903739, 7.39670640039587408191046820043
