| L(s) = 1 | + 3-s + 5-s − 7-s + 9-s − 13-s + 15-s + 2·17-s + 4·19-s − 21-s + 25-s + 27-s − 6·29-s + 8·31-s − 35-s − 2·37-s − 39-s − 10·41-s − 8·43-s + 45-s + 12·47-s + 49-s + 2·51-s + 6·53-s + 4·57-s − 4·59-s − 6·61-s − 63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.277·13-s + 0.258·15-s + 0.485·17-s + 0.917·19-s − 0.218·21-s + 1/5·25-s + 0.192·27-s − 1.11·29-s + 1.43·31-s − 0.169·35-s − 0.328·37-s − 0.160·39-s − 1.56·41-s − 1.21·43-s + 0.149·45-s + 1.75·47-s + 1/7·49-s + 0.280·51-s + 0.824·53-s + 0.529·57-s − 0.520·59-s − 0.768·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 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
−13.98107210964243, −13.65681097256901, −13.37662786253972, −12.70915287711535, −12.12896010774815, −11.85900932494869, −11.19397294438932, −10.47307189287922, −9.997125836963752, −9.822243728898674, −9.056095578490012, −8.778070550401240, −8.077770360189565, −7.589206666513063, −7.021732649300498, −6.632011241933532, −5.779953920386768, −5.508909005281870, −4.761798521717656, −4.198084317080366, −3.395995320064449, −3.073246320613343, −2.392268547089127, −1.668789723489094, −1.047539627730551, 0,
1.047539627730551, 1.668789723489094, 2.392268547089127, 3.073246320613343, 3.395995320064449, 4.198084317080366, 4.761798521717656, 5.508909005281870, 5.779953920386768, 6.632011241933532, 7.021732649300498, 7.589206666513063, 8.077770360189565, 8.778070550401240, 9.056095578490012, 9.822243728898674, 9.997125836963752, 10.47307189287922, 11.19397294438932, 11.85900932494869, 12.12896010774815, 12.70915287711535, 13.37662786253972, 13.65681097256901, 13.98107210964243
