| L(s) = 1 | − 4·7-s + 4·11-s + 6·13-s + 2·17-s + 5·19-s − 6·23-s − 4·29-s + 4·31-s − 8·37-s + 7·41-s + 5·43-s + 10·47-s + 9·49-s + 2·53-s − 3·59-s − 4·61-s + 9·67-s − 12·71-s − 7·73-s − 16·77-s + 10·79-s − 7·83-s − 15·89-s − 24·91-s + 11·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 1.20·11-s + 1.66·13-s + 0.485·17-s + 1.14·19-s − 1.25·23-s − 0.742·29-s + 0.718·31-s − 1.31·37-s + 1.09·41-s + 0.762·43-s + 1.45·47-s + 9/7·49-s + 0.274·53-s − 0.390·59-s − 0.512·61-s + 1.09·67-s − 1.42·71-s − 0.819·73-s − 1.82·77-s + 1.12·79-s − 0.768·83-s − 1.58·89-s − 2.51·91-s + 1.11·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.524432235\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.524432235\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 7 T + p T^{2} \) | 1.41.ah |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - 9 T + p T^{2} \) | 1.67.aj |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 7 T + p T^{2} \) | 1.83.h |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 - 11 T + p T^{2} \) | 1.97.al |
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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
−14.08722314424834, −13.72055788108794, −13.34689943727465, −12.68450809885371, −12.08463130010570, −11.95996543640932, −11.15655386182686, −10.69237810673357, −10.06290112795315, −9.577291327508589, −9.201973767829266, −8.693162650287712, −8.110045083650147, −7.293259861731275, −6.982596323672254, −6.204377214112507, −5.912136087549736, −5.573434136793301, −4.342642694461636, −3.957590690202000, −3.389855423873257, −3.025927517550263, −1.995892626609413, −1.211957023056056, −0.5976172137726586,
0.5976172137726586, 1.211957023056056, 1.995892626609413, 3.025927517550263, 3.389855423873257, 3.957590690202000, 4.342642694461636, 5.573434136793301, 5.912136087549736, 6.204377214112507, 6.982596323672254, 7.293259861731275, 8.110045083650147, 8.693162650287712, 9.201973767829266, 9.577291327508589, 10.06290112795315, 10.69237810673357, 11.15655386182686, 11.95996543640932, 12.08463130010570, 12.68450809885371, 13.34689943727465, 13.72055788108794, 14.08722314424834
