| L(s) = 1 | + 3-s − 4·5-s + 2·7-s + 9-s − 4·13-s − 4·15-s − 2·17-s − 4·19-s + 2·21-s + 6·23-s + 11·25-s + 27-s + 6·29-s + 4·31-s − 8·35-s − 2·37-s − 4·39-s − 6·41-s − 8·43-s − 4·45-s + 10·47-s − 3·49-s − 2·51-s − 12·53-s − 4·57-s − 8·61-s + 2·63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.78·5-s + 0.755·7-s + 1/3·9-s − 1.10·13-s − 1.03·15-s − 0.485·17-s − 0.917·19-s + 0.436·21-s + 1.25·23-s + 11/5·25-s + 0.192·27-s + 1.11·29-s + 0.718·31-s − 1.35·35-s − 0.328·37-s − 0.640·39-s − 0.937·41-s − 1.21·43-s − 0.596·45-s + 1.45·47-s − 3/7·49-s − 0.280·51-s − 1.64·53-s − 0.529·57-s − 1.02·61-s + 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.400830416\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.400830416\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−16.36305162383938, −15.56190364842451, −15.19659908270633, −15.00498698521983, −14.24141743033499, −13.74488951955920, −12.76619827493770, −12.45061908497672, −11.80170427199273, −11.32771709844640, −10.71664079855800, −10.15046442800272, −9.185362584436592, −8.619798145924143, −8.117661968180472, −7.717682173615206, −6.943777513611653, −6.596765531955217, −5.139201143139340, −4.632339921460955, −4.259838044433443, −3.277799823668986, −2.757628261145636, −1.700894543846752, −0.5234190105425316,
0.5234190105425316, 1.700894543846752, 2.757628261145636, 3.277799823668986, 4.259838044433443, 4.632339921460955, 5.139201143139340, 6.596765531955217, 6.943777513611653, 7.717682173615206, 8.117661968180472, 8.619798145924143, 9.185362584436592, 10.15046442800272, 10.71664079855800, 11.32771709844640, 11.80170427199273, 12.45061908497672, 12.76619827493770, 13.74488951955920, 14.24141743033499, 15.00498698521983, 15.19659908270633, 15.56190364842451, 16.36305162383938
