| L(s) = 1 | + 3-s + 5-s + 9-s − 6·11-s − 13-s + 15-s + 4·23-s + 25-s + 27-s − 2·29-s + 6·31-s − 6·33-s − 6·37-s − 39-s − 2·41-s − 10·43-s + 45-s + 6·47-s − 7·49-s − 6·53-s − 6·55-s + 6·61-s − 65-s − 2·67-s + 4·69-s − 10·71-s + 2·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.80·11-s − 0.277·13-s + 0.258·15-s + 0.834·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 1.07·31-s − 1.04·33-s − 0.986·37-s − 0.160·39-s − 0.312·41-s − 1.52·43-s + 0.149·45-s + 0.875·47-s − 49-s − 0.824·53-s − 0.809·55-s + 0.768·61-s − 0.124·65-s − 0.244·67-s + 0.481·69-s − 1.18·71-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.589458536\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.589458536\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 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 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 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
−12.94828819015359, −12.71207604408756, −12.07470587412528, −11.59800490297426, −10.85338796444402, −10.65737978992675, −10.05728666975964, −9.767381794137009, −9.257570124216181, −8.582286579781668, −8.256305149037199, −7.876903930634108, −7.213702131185395, −6.884043084114761, −6.291471668325385, −5.554943142430070, −5.200151298398994, −4.796591888476896, −4.177325849075263, −3.373169381508754, −2.907682910589747, −2.570532746917008, −1.865399511379600, −1.315411101161123, −0.3206748547502533,
0.3206748547502533, 1.315411101161123, 1.865399511379600, 2.570532746917008, 2.907682910589747, 3.373169381508754, 4.177325849075263, 4.796591888476896, 5.200151298398994, 5.554943142430070, 6.291471668325385, 6.884043084114761, 7.213702131185395, 7.876903930634108, 8.256305149037199, 8.582286579781668, 9.257570124216181, 9.767381794137009, 10.05728666975964, 10.65737978992675, 10.85338796444402, 11.59800490297426, 12.07470587412528, 12.71207604408756, 12.94828819015359
