| L(s) = 1 | − 2·5-s + 7-s − 2·11-s + 2·13-s − 6·17-s + 6·19-s − 6·23-s − 25-s − 9·29-s − 2·31-s − 2·35-s − 9·37-s + 10·41-s − 2·43-s + 10·47-s − 6·49-s + 8·53-s + 4·55-s − 3·59-s + 6·61-s − 4·65-s + 10·67-s − 5·71-s − 9·73-s − 2·77-s − 12·79-s − 6·83-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s − 0.603·11-s + 0.554·13-s − 1.45·17-s + 1.37·19-s − 1.25·23-s − 1/5·25-s − 1.67·29-s − 0.359·31-s − 0.338·35-s − 1.47·37-s + 1.56·41-s − 0.304·43-s + 1.45·47-s − 6/7·49-s + 1.09·53-s + 0.539·55-s − 0.390·59-s + 0.768·61-s − 0.496·65-s + 1.22·67-s − 0.593·71-s − 1.05·73-s − 0.227·77-s − 1.35·79-s − 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4165478170\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4165478170\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 397 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 9 T + p T^{2} \) | 1.37.j |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−13.08784120085182, −12.29092368126536, −12.05650194135229, −11.41264873996336, −11.13879916281958, −10.85495754512035, −10.11454866219911, −9.704332867591046, −9.115088710238326, −8.604846846616817, −8.272660260116752, −7.662218915390133, −7.265869646003050, −7.022100950275347, −6.110337128039337, −5.628948214112797, −5.342816811220284, −4.498954114081779, −4.105612446070550, −3.709360640362426, −3.100380796870248, −2.335571890770612, −1.881033523609968, −1.127723881879502, −0.1864892896122346,
0.1864892896122346, 1.127723881879502, 1.881033523609968, 2.335571890770612, 3.100380796870248, 3.709360640362426, 4.105612446070550, 4.498954114081779, 5.342816811220284, 5.628948214112797, 6.110337128039337, 7.022100950275347, 7.265869646003050, 7.662218915390133, 8.272660260116752, 8.604846846616817, 9.115088710238326, 9.704332867591046, 10.11454866219911, 10.85495754512035, 11.13879916281958, 11.41264873996336, 12.05650194135229, 12.29092368126536, 13.08784120085182
