| L(s) = 1 | + 2-s + 4-s + 5-s + 4·7-s + 8-s + 10-s + 4·11-s − 5·13-s + 4·14-s + 16-s − 3·19-s + 20-s + 4·22-s − 8·23-s − 4·25-s − 5·26-s + 4·28-s − 7·31-s + 32-s + 4·35-s + 4·37-s − 3·38-s + 40-s − 41-s − 6·43-s + 4·44-s − 8·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 1.51·7-s + 0.353·8-s + 0.316·10-s + 1.20·11-s − 1.38·13-s + 1.06·14-s + 1/4·16-s − 0.688·19-s + 0.223·20-s + 0.852·22-s − 1.66·23-s − 4/5·25-s − 0.980·26-s + 0.755·28-s − 1.25·31-s + 0.176·32-s + 0.676·35-s + 0.657·37-s − 0.486·38-s + 0.158·40-s − 0.156·41-s − 0.914·43-s + 0.603·44-s − 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 213282 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 213282 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.928072602\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.928072602\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 17 | \( 1 \) | |
| 41 | \( 1 + T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 - 7 T + p T^{2} \) | 1.59.ah |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 + 7 T + p T^{2} \) | 1.71.h |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 - 15 T + p T^{2} \) | 1.83.ap |
| 89 | \( 1 + 13 T + p T^{2} \) | 1.89.n |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−12.94939255519536, −12.49404093856213, −11.99817149279028, −11.71547774992131, −11.32326732026999, −10.75100343642929, −10.27775215456975, −9.689350875891135, −9.375925926078703, −8.704326791916791, −8.122251825802809, −7.757084453420524, −7.296904113222435, −6.656791259092310, −6.231632962902982, −5.593900624477163, −5.280962632799135, −4.617732446703791, −4.211490516235148, −3.864899611786365, −3.052565537442114, −2.157952673549272, −1.967598877900274, −1.536810160730445, −0.5035198731693473,
0.5035198731693473, 1.536810160730445, 1.967598877900274, 2.157952673549272, 3.052565537442114, 3.864899611786365, 4.211490516235148, 4.617732446703791, 5.280962632799135, 5.593900624477163, 6.231632962902982, 6.656791259092310, 7.296904113222435, 7.757084453420524, 8.122251825802809, 8.704326791916791, 9.375925926078703, 9.689350875891135, 10.27775215456975, 10.75100343642929, 11.32326732026999, 11.71547774992131, 11.99817149279028, 12.49404093856213, 12.94939255519536
