| L(s) = 1 | + 3·3-s + 2·5-s − 7-s + 6·9-s + 2·11-s − 13-s + 6·15-s + 2·19-s − 3·21-s − 23-s − 25-s + 9·27-s − 3·29-s − 31-s + 6·33-s − 2·35-s − 2·37-s − 3·39-s − 41-s + 8·43-s + 12·45-s − 5·47-s + 49-s − 6·53-s + 4·55-s + 6·57-s + 6·61-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.894·5-s − 0.377·7-s + 2·9-s + 0.603·11-s − 0.277·13-s + 1.54·15-s + 0.458·19-s − 0.654·21-s − 0.208·23-s − 1/5·25-s + 1.73·27-s − 0.557·29-s − 0.179·31-s + 1.04·33-s − 0.338·35-s − 0.328·37-s − 0.480·39-s − 0.156·41-s + 1.21·43-s + 1.78·45-s − 0.729·47-s + 1/7·49-s − 0.824·53-s + 0.539·55-s + 0.794·57-s + 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.484992687\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.484992687\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 23 | \( 1 + T \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + T + p T^{2} \) | 1.41.b |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 5 T + p T^{2} \) | 1.47.f |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 - 7 T + p T^{2} \) | 1.71.ah |
| 73 | \( 1 - 13 T + p T^{2} \) | 1.73.an |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 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
−9.579281767076668646883614144046, −9.003560197720035813565931085517, −8.151484450294933667121087039727, −7.35717033014221846933977304477, −6.53048822074249301258004827687, −5.48645111741811701110312814163, −4.21230810906643869776513722821, −3.36833968235597921897432637920, −2.44863282120741202902662275901, −1.56670037343810394124028039892,
1.56670037343810394124028039892, 2.44863282120741202902662275901, 3.36833968235597921897432637920, 4.21230810906643869776513722821, 5.48645111741811701110312814163, 6.53048822074249301258004827687, 7.35717033014221846933977304477, 8.151484450294933667121087039727, 9.003560197720035813565931085517, 9.579281767076668646883614144046
