| L(s) = 1 | − 5-s − 2·7-s + 2·11-s − 4·13-s + 2·17-s − 19-s + 25-s + 4·29-s + 8·31-s + 2·35-s − 8·37-s + 12·41-s − 6·43-s − 3·49-s + 2·53-s − 2·55-s + 2·61-s + 4·65-s − 8·67-s − 8·71-s + 6·73-s − 4·77-s + 8·79-s + 8·83-s − 2·85-s + 4·89-s + 8·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s + 0.603·11-s − 1.10·13-s + 0.485·17-s − 0.229·19-s + 1/5·25-s + 0.742·29-s + 1.43·31-s + 0.338·35-s − 1.31·37-s + 1.87·41-s − 0.914·43-s − 3/7·49-s + 0.274·53-s − 0.269·55-s + 0.256·61-s + 0.496·65-s − 0.977·67-s − 0.949·71-s + 0.702·73-s − 0.455·77-s + 0.900·79-s + 0.878·83-s − 0.216·85-s + 0.423·89-s + 0.838·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 19 | \( 1 + T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−7.64319757853443199958034364576, −6.79552101067111065865489660245, −6.44204088330689517018139465192, −5.46673843789851780699537741895, −4.69176338260732513073358742865, −3.98675979114130661644889612077, −3.14091383291310148117946089007, −2.45915554727424092767865010155, −1.17092340766453562714934748288, 0,
1.17092340766453562714934748288, 2.45915554727424092767865010155, 3.14091383291310148117946089007, 3.98675979114130661644889612077, 4.69176338260732513073358742865, 5.46673843789851780699537741895, 6.44204088330689517018139465192, 6.79552101067111065865489660245, 7.64319757853443199958034364576
