| L(s) = 1 | + 3-s + 7-s + 9-s + 2·11-s + 2·13-s − 17-s + 21-s − 4·23-s + 27-s − 2·29-s − 2·31-s + 2·33-s + 10·37-s + 2·39-s + 10·41-s + 10·43-s + 2·47-s + 49-s − 51-s + 2·53-s − 8·59-s + 6·61-s + 63-s − 6·67-s − 4·69-s + 6·71-s − 6·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.603·11-s + 0.554·13-s − 0.242·17-s + 0.218·21-s − 0.834·23-s + 0.192·27-s − 0.371·29-s − 0.359·31-s + 0.348·33-s + 1.64·37-s + 0.320·39-s + 1.56·41-s + 1.52·43-s + 0.291·47-s + 1/7·49-s − 0.140·51-s + 0.274·53-s − 1.04·59-s + 0.768·61-s + 0.125·63-s − 0.733·67-s − 0.481·69-s + 0.712·71-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.607756228\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.607756228\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 + T \) | |
| good | 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 6 T + p T^{2} \) | 1.67.g |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−14.78891435426205, −14.40022031432353, −13.98538524316629, −13.41800131009536, −12.85450028926676, −12.39238939310275, −11.74184438408250, −11.15161972467490, −10.81560885440072, −10.07606102793122, −9.415695652362094, −9.094021487980475, −8.528938015136075, −7.731485736747819, −7.622869178934187, −6.746123634220432, −6.059769544042019, −5.720235411238006, −4.735140558073371, −4.105822009716103, −3.817772523985359, −2.828235930587855, −2.269600398753690, −1.474216008139107, −0.7142044841829094,
0.7142044841829094, 1.474216008139107, 2.269600398753690, 2.828235930587855, 3.817772523985359, 4.105822009716103, 4.735140558073371, 5.720235411238006, 6.059769544042019, 6.746123634220432, 7.622869178934187, 7.731485736747819, 8.528938015136075, 9.094021487980475, 9.415695652362094, 10.07606102793122, 10.81560885440072, 11.15161972467490, 11.74184438408250, 12.39238939310275, 12.85450028926676, 13.41800131009536, 13.98538524316629, 14.40022031432353, 14.78891435426205
