| L(s) = 1 | − 7-s − 6·11-s + 4·13-s − 17-s − 8·19-s + 8·23-s − 6·31-s + 6·37-s − 2·41-s + 8·43-s + 8·47-s + 49-s + 4·53-s + 4·59-s − 12·61-s + 8·67-s + 14·71-s + 14·73-s + 6·77-s + 2·89-s − 4·91-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 1.80·11-s + 1.10·13-s − 0.242·17-s − 1.83·19-s + 1.66·23-s − 1.07·31-s + 0.986·37-s − 0.312·41-s + 1.21·43-s + 1.16·47-s + 1/7·49-s + 0.549·53-s + 0.520·59-s − 1.53·61-s + 0.977·67-s + 1.66·71-s + 1.63·73-s + 0.683·77-s + 0.211·89-s − 0.419·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 214200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 214200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.768552157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.768552157\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 + T \) | |
| good | 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−12.93786987928912, −12.60532341596092, −12.40442394381823, −11.27237983554425, −11.05616815783323, −10.75093457761627, −10.38336817944943, −9.714698270281916, −9.142982364190280, −8.726107781572894, −8.367261989282525, −7.699795243382586, −7.411040311460587, −6.622730775308651, −6.379632288686426, −5.664564403327097, −5.319627768297619, −4.731185558976404, −4.100938406994377, −3.649396518279897, −2.935594855323623, −2.433544394375604, −2.020184284377528, −0.9998601858854212, −0.4211100018061133,
0.4211100018061133, 0.9998601858854212, 2.020184284377528, 2.433544394375604, 2.935594855323623, 3.649396518279897, 4.100938406994377, 4.731185558976404, 5.319627768297619, 5.664564403327097, 6.379632288686426, 6.622730775308651, 7.411040311460587, 7.699795243382586, 8.367261989282525, 8.726107781572894, 9.142982364190280, 9.714698270281916, 10.38336817944943, 10.75093457761627, 11.05616815783323, 11.27237983554425, 12.40442394381823, 12.60532341596092, 12.93786987928912
