| L(s) = 1 | − 5·7-s + 7·13-s + 8·19-s + 11·31-s + 10·37-s − 8·43-s + 18·49-s − 61-s + 16·67-s − 17·73-s + 17·79-s − 35·91-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.88·7-s + 1.94·13-s + 1.83·19-s + 1.97·31-s + 1.64·37-s − 1.21·43-s + 18/7·49-s − 0.128·61-s + 1.95·67-s − 1.98·73-s + 1.91·79-s − 3.66·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.230438516\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.230438516\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 5 T + p T^{2} \) | 1.7.f |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 7 T + p T^{2} \) | 1.13.ah |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 11 T + p T^{2} \) | 1.31.al |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 16 T + p T^{2} \) | 1.67.aq |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 17 T + p T^{2} \) | 1.73.r |
| 79 | \( 1 - 17 T + p T^{2} \) | 1.79.ar |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−15.55692475151466, −15.10289177800695, −14.07495240714305, −13.66029499640959, −13.32577968892047, −12.87436036816736, −12.15527753793483, −11.63120780845690, −11.12894324854242, −10.34839799744757, −9.826904629879057, −9.514924010464926, −8.825672381750354, −8.242719292248281, −7.592142973024989, −6.768255188315408, −6.368034932589662, −5.943256477108168, −5.251390599820795, −4.284686433756593, −3.587990523824895, −3.180034309028929, −2.576647076558712, −1.240086962146694, −0.6822123212323219,
0.6822123212323219, 1.240086962146694, 2.576647076558712, 3.180034309028929, 3.587990523824895, 4.284686433756593, 5.251390599820795, 5.943256477108168, 6.368034932589662, 6.768255188315408, 7.592142973024989, 8.242719292248281, 8.825672381750354, 9.514924010464926, 9.826904629879057, 10.34839799744757, 11.12894324854242, 11.63120780845690, 12.15527753793483, 12.87436036816736, 13.32577968892047, 13.66029499640959, 14.07495240714305, 15.10289177800695, 15.55692475151466
