| L(s) = 1 | + 3-s − 5-s + 9-s − 5·11-s − 13-s − 15-s + 3·17-s − 19-s + 3·23-s − 4·25-s + 27-s − 9·29-s − 4·31-s − 5·33-s + 11·37-s − 39-s + 5·43-s − 45-s + 8·47-s + 3·51-s + 2·53-s + 5·55-s − 57-s + 4·59-s − 15·61-s + 65-s + 2·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.50·11-s − 0.277·13-s − 0.258·15-s + 0.727·17-s − 0.229·19-s + 0.625·23-s − 4/5·25-s + 0.192·27-s − 1.67·29-s − 0.718·31-s − 0.870·33-s + 1.80·37-s − 0.160·39-s + 0.762·43-s − 0.149·45-s + 1.16·47-s + 0.420·51-s + 0.274·53-s + 0.674·55-s − 0.132·57-s + 0.520·59-s − 1.92·61-s + 0.124·65-s + 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 11 T + p T^{2} \) | 1.37.al |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 15 T + p T^{2} \) | 1.61.p |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−13.64761739548817, −13.31611338191691, −12.81676320237815, −12.55092240240313, −11.86771725137404, −11.29569735441757, −10.91334904045580, −10.35660314779614, −9.911497547675453, −9.350410948651421, −8.923316275261747, −8.328707980695236, −7.671208331576566, −7.490856146702189, −7.252262401012287, −6.122095325566035, −5.745496996679953, −5.276984904196111, −4.481420230144556, −4.149351078292221, −3.367465981641054, −2.932370203638113, −2.319834730239857, −1.745825855423533, −0.7732808822490456, 0,
0.7732808822490456, 1.745825855423533, 2.319834730239857, 2.932370203638113, 3.367465981641054, 4.149351078292221, 4.481420230144556, 5.276984904196111, 5.745496996679953, 6.122095325566035, 7.252262401012287, 7.490856146702189, 7.671208331576566, 8.328707980695236, 8.923316275261747, 9.350410948651421, 9.911497547675453, 10.35660314779614, 10.91334904045580, 11.29569735441757, 11.86771725137404, 12.55092240240313, 12.81676320237815, 13.31611338191691, 13.64761739548817
