| L(s) = 1 | − 2-s + 4-s − 3·5-s − 8-s + 3·10-s − 3·13-s + 16-s − 3·17-s + 6·19-s − 3·20-s + 3·23-s + 4·25-s + 3·26-s − 6·29-s + 4·31-s − 32-s + 3·34-s − 2·37-s − 6·38-s + 3·40-s + 3·41-s + 6·43-s − 3·46-s − 7·49-s − 4·50-s − 3·52-s + 9·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.34·5-s − 0.353·8-s + 0.948·10-s − 0.832·13-s + 1/4·16-s − 0.727·17-s + 1.37·19-s − 0.670·20-s + 0.625·23-s + 4/5·25-s + 0.588·26-s − 1.11·29-s + 0.718·31-s − 0.176·32-s + 0.514·34-s − 0.328·37-s − 0.973·38-s + 0.474·40-s + 0.468·41-s + 0.914·43-s − 0.442·46-s − 49-s − 0.565·50-s − 0.416·52-s + 1.23·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 15 T + p T^{2} \) | 1.61.ap |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 15 T + p T^{2} \) | 1.79.p |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 17 T + p T^{2} \) | 1.97.ar |
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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
−7.46354765247809194364825745084, −7.39663128358826465365171179306, −6.52135776067638513573341137206, −5.49770113895846312158943920178, −4.74776542095034696056860831171, −3.91568859743100595773186778161, −3.15034774605527086362874480721, −2.29348062381161737795732444319, −1.00724616154225706141437973244, 0,
1.00724616154225706141437973244, 2.29348062381161737795732444319, 3.15034774605527086362874480721, 3.91568859743100595773186778161, 4.74776542095034696056860831171, 5.49770113895846312158943920178, 6.52135776067638513573341137206, 7.39663128358826465365171179306, 7.46354765247809194364825745084
