| L(s) = 1 | + 2·3-s − 2·5-s − 2·7-s + 9-s − 4·15-s − 6·17-s + 19-s − 4·21-s − 25-s − 4·27-s − 4·29-s + 8·31-s + 4·35-s − 6·37-s + 10·41-s − 4·43-s − 2·45-s + 2·47-s − 3·49-s − 12·51-s + 12·53-s + 2·57-s − 4·59-s − 6·61-s − 2·63-s + 4·67-s − 8·71-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s − 0.755·7-s + 1/3·9-s − 1.03·15-s − 1.45·17-s + 0.229·19-s − 0.872·21-s − 1/5·25-s − 0.769·27-s − 0.742·29-s + 1.43·31-s + 0.676·35-s − 0.986·37-s + 1.56·41-s − 0.609·43-s − 0.298·45-s + 0.291·47-s − 3/7·49-s − 1.68·51-s + 1.64·53-s + 0.264·57-s − 0.520·59-s − 0.768·61-s − 0.251·63-s + 0.488·67-s − 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 205504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 205504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| 19 | \( 1 - T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−13.28063351056298, −12.99597120830773, −12.39007205546872, −11.70761905787681, −11.62488227509829, −10.91649583160267, −10.43220524803085, −9.861735070015966, −9.389392340722810, −8.925557909551517, −8.611233047507435, −8.069443300796194, −7.663372348063168, −7.083899332378306, −6.779475171883811, −6.007246096238340, −5.654996291783999, −4.707506953470900, −4.269661628669113, −3.876448294589897, −3.206216515129353, −2.914994207728787, −2.251265303550303, −1.718429962826116, −0.6621659193602437, 0,
0.6621659193602437, 1.718429962826116, 2.251265303550303, 2.914994207728787, 3.206216515129353, 3.876448294589897, 4.269661628669113, 4.707506953470900, 5.654996291783999, 6.007246096238340, 6.779475171883811, 7.083899332378306, 7.663372348063168, 8.069443300796194, 8.611233047507435, 8.925557909551517, 9.389392340722810, 9.861735070015966, 10.43220524803085, 10.91649583160267, 11.62488227509829, 11.70761905787681, 12.39007205546872, 12.99597120830773, 13.28063351056298
