| L(s) = 1 | − 2·5-s + 2·7-s + 4·11-s − 4·13-s + 7·17-s − 3·19-s − 25-s − 4·29-s + 6·31-s − 4·35-s − 2·37-s − 6·41-s − 5·43-s + 10·47-s − 3·49-s − 8·55-s − 5·59-s + 4·61-s + 8·65-s − 5·67-s − 14·71-s + 15·73-s + 8·77-s − 12·79-s − 15·83-s − 14·85-s + 10·89-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.755·7-s + 1.20·11-s − 1.10·13-s + 1.69·17-s − 0.688·19-s − 1/5·25-s − 0.742·29-s + 1.07·31-s − 0.676·35-s − 0.328·37-s − 0.937·41-s − 0.762·43-s + 1.45·47-s − 3/7·49-s − 1.07·55-s − 0.650·59-s + 0.512·61-s + 0.992·65-s − 0.610·67-s − 1.66·71-s + 1.75·73-s + 0.911·77-s − 1.35·79-s − 1.64·83-s − 1.51·85-s + 1.05·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 5 T + p T^{2} \) | 1.59.f |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 - 15 T + p T^{2} \) | 1.73.ap |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 15 T + p T^{2} \) | 1.83.p |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 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
−12.60479765474089, −12.41246278586411, −11.83378794863737, −11.63938686626057, −11.37105354477130, −10.47463695356465, −10.24549264362225, −9.744711228759090, −9.188310200964534, −8.691829998531518, −8.221902305391703, −7.770311812235763, −7.413516458613458, −6.978992064349427, −6.358260458581650, −5.825594129474283, −5.256235632100351, −4.752392281166120, −4.282800985848664, −3.814324536308554, −3.307079402093842, −2.733487960272792, −1.902815548156939, −1.478084866248156, −0.7650726055067931, 0,
0.7650726055067931, 1.478084866248156, 1.902815548156939, 2.733487960272792, 3.307079402093842, 3.814324536308554, 4.282800985848664, 4.752392281166120, 5.256235632100351, 5.825594129474283, 6.358260458581650, 6.978992064349427, 7.413516458613458, 7.770311812235763, 8.221902305391703, 8.691829998531518, 9.188310200964534, 9.744711228759090, 10.24549264362225, 10.47463695356465, 11.37105354477130, 11.63938686626057, 11.83378794863737, 12.41246278586411, 12.60479765474089
