| L(s) = 1 | − 3-s + 2·7-s + 9-s + 2·11-s + 13-s + 17-s − 6·19-s − 2·21-s − 4·23-s − 27-s + 2·29-s − 2·31-s − 2·33-s + 2·37-s − 39-s − 10·41-s − 8·43-s + 6·47-s − 3·49-s − 51-s + 2·53-s + 6·57-s + 6·59-s − 2·61-s + 2·63-s + 2·67-s + 4·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s + 0.603·11-s + 0.277·13-s + 0.242·17-s − 1.37·19-s − 0.436·21-s − 0.834·23-s − 0.192·27-s + 0.371·29-s − 0.359·31-s − 0.348·33-s + 0.328·37-s − 0.160·39-s − 1.56·41-s − 1.21·43-s + 0.875·47-s − 3/7·49-s − 0.140·51-s + 0.274·53-s + 0.794·57-s + 0.781·59-s − 0.256·61-s + 0.251·63-s + 0.244·67-s + 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 265200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 265200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.358753209\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.358753209\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| 17 | \( 1 - T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−12.84735996208925, −12.13140230068858, −11.86063646849205, −11.54806130200309, −10.86326106361188, −10.66916599062410, −10.02683354901330, −9.755591077892861, −8.923973093534444, −8.654020937305996, −8.091037005067747, −7.775812065988636, −6.969738211864564, −6.633549438157532, −6.236835879348795, −5.585800558748797, −5.201621257369446, −4.594607543662985, −4.121993100999573, −3.731585056447882, −2.991837135895331, −2.191928919852339, −1.713561774176855, −1.214747688968773, −0.3342347438998186,
0.3342347438998186, 1.214747688968773, 1.713561774176855, 2.191928919852339, 2.991837135895331, 3.731585056447882, 4.121993100999573, 4.594607543662985, 5.201621257369446, 5.585800558748797, 6.236835879348795, 6.633549438157532, 6.969738211864564, 7.775812065988636, 8.091037005067747, 8.654020937305996, 8.923973093534444, 9.755591077892861, 10.02683354901330, 10.66916599062410, 10.86326106361188, 11.54806130200309, 11.86063646849205, 12.13140230068858, 12.84735996208925
