| L(s) = 1 | + 2-s − 2·3-s + 4-s − 5-s − 2·6-s − 7-s + 8-s + 9-s − 10-s + 11-s − 2·12-s + 4·13-s − 14-s + 2·15-s + 16-s + 18-s − 20-s + 2·21-s + 22-s − 4·23-s − 2·24-s + 25-s + 4·26-s + 4·27-s − 28-s + 6·29-s + 2·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.447·5-s − 0.816·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s − 0.577·12-s + 1.10·13-s − 0.267·14-s + 0.516·15-s + 1/4·16-s + 0.235·18-s − 0.223·20-s + 0.436·21-s + 0.213·22-s − 0.834·23-s − 0.408·24-s + 1/5·25-s + 0.784·26-s + 0.769·27-s − 0.188·28-s + 1.11·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 277970 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277970 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.994668865\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.994668865\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| 19 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−12.81207380052812, −12.07408849161161, −11.91848183994140, −11.49303640241506, −10.95889332673736, −10.81002555958597, −9.969802944614287, −9.889514319404926, −8.962555637683363, −8.490577570370128, −8.139527537913782, −7.404212440837617, −6.922737188120771, −6.429693451654101, −6.144493621687490, −5.634293984536976, −5.227027252098447, −4.553368369183975, −4.179823372879667, −3.598596995753034, −3.165020670411760, −2.472326403066069, −1.694905033107491, −1.036523577689552, −0.4125197161580008,
0.4125197161580008, 1.036523577689552, 1.694905033107491, 2.472326403066069, 3.165020670411760, 3.598596995753034, 4.179823372879667, 4.553368369183975, 5.227027252098447, 5.634293984536976, 6.144493621687490, 6.429693451654101, 6.922737188120771, 7.404212440837617, 8.139527537913782, 8.490577570370128, 8.962555637683363, 9.889514319404926, 9.969802944614287, 10.81002555958597, 10.95889332673736, 11.49303640241506, 11.91848183994140, 12.07408849161161, 12.81207380052812
