| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 8-s + 9-s + 10-s + 4·11-s − 12-s + 15-s + 16-s − 18-s − 5·19-s − 20-s − 4·22-s + 23-s + 24-s + 25-s − 27-s − 4·29-s − 30-s + 6·31-s − 32-s − 4·33-s + 36-s − 7·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s − 0.288·12-s + 0.258·15-s + 1/4·16-s − 0.235·18-s − 1.14·19-s − 0.223·20-s − 0.852·22-s + 0.208·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s − 0.742·29-s − 0.182·30-s + 1.07·31-s − 0.176·32-s − 0.696·33-s + 1/6·36-s − 1.15·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - T + p T^{2} \) | 1.47.ab |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−13.17315173566702, −12.36535629880999, −11.95365081868061, −11.79345018745775, −11.20547091708782, −10.81702868299395, −10.29556887670343, −9.874988771419232, −9.451475584710963, −8.694594463034696, −8.527447243024864, −8.127289497709744, −7.231925528078646, −6.967880747260061, −6.599647552050914, −6.118784760204529, −5.417883679182886, −5.035949666172143, −4.147212871541098, −3.999049638573440, −3.304989633063054, −2.602236736759905, −1.844064950835170, −1.417173372763242, −0.6628028743889272, 0,
0.6628028743889272, 1.417173372763242, 1.844064950835170, 2.602236736759905, 3.304989633063054, 3.999049638573440, 4.147212871541098, 5.035949666172143, 5.417883679182886, 6.118784760204529, 6.599647552050914, 6.967880747260061, 7.231925528078646, 8.127289497709744, 8.527447243024864, 8.694594463034696, 9.451475584710963, 9.874988771419232, 10.29556887670343, 10.81702868299395, 11.20547091708782, 11.79345018745775, 11.95365081868061, 12.36535629880999, 13.17315173566702
