| L(s) = 1 | + 2-s − 3-s + 4-s − 2·5-s − 6-s + 2·7-s + 8-s + 9-s − 2·10-s + 2·11-s − 12-s + 2·14-s + 2·15-s + 16-s − 4·17-s + 18-s + 19-s − 2·20-s − 2·21-s + 2·22-s + 23-s − 24-s − 25-s − 27-s + 2·28-s + 3·29-s + 2·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.603·11-s − 0.288·12-s + 0.534·14-s + 0.516·15-s + 1/4·16-s − 0.970·17-s + 0.235·18-s + 0.229·19-s − 0.447·20-s − 0.436·21-s + 0.426·22-s + 0.208·23-s − 0.204·24-s − 1/5·25-s − 0.192·27-s + 0.377·28-s + 0.557·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 255162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 255162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 23 | \( 1 - T \) | |
| 43 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - T + p T^{2} \) | 1.41.ab |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 13 T + p T^{2} \) | 1.53.an |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 - 10 T + p T^{2} \) | 1.83.ak |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
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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
−13.00297695387253, −12.49748206858760, −12.05102524224084, −11.69123617522923, −11.38383116117174, −10.87523069675345, −10.65880096704522, −9.865930064605720, −9.404283794368821, −8.754339555053115, −8.353539703165871, −7.743584327229859, −7.362390829077324, −6.939198475919724, −6.268566753125392, −6.014414550316739, −5.255513681090712, −4.843060926907544, −4.361258633292731, −3.965733999692807, −3.530363470622491, −2.666554857480837, −2.203343414621099, −1.401152398199126, −0.8659579088702893, 0,
0.8659579088702893, 1.401152398199126, 2.203343414621099, 2.666554857480837, 3.530363470622491, 3.965733999692807, 4.361258633292731, 4.843060926907544, 5.255513681090712, 6.014414550316739, 6.268566753125392, 6.939198475919724, 7.362390829077324, 7.743584327229859, 8.353539703165871, 8.754339555053115, 9.404283794368821, 9.865930064605720, 10.65880096704522, 10.87523069675345, 11.38383116117174, 11.69123617522923, 12.05102524224084, 12.49748206858760, 13.00297695387253
