| L(s) = 1 | − 5-s − 4·7-s + 2·11-s + 13-s − 2·17-s + 6·19-s + 6·23-s + 25-s + 2·29-s − 10·31-s + 4·35-s + 2·37-s + 6·41-s − 10·43-s − 4·47-s + 9·49-s + 2·53-s − 2·55-s + 6·59-s − 2·61-s − 65-s + 4·67-s − 6·71-s − 6·73-s − 8·77-s − 12·79-s − 16·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.51·7-s + 0.603·11-s + 0.277·13-s − 0.485·17-s + 1.37·19-s + 1.25·23-s + 1/5·25-s + 0.371·29-s − 1.79·31-s + 0.676·35-s + 0.328·37-s + 0.937·41-s − 1.52·43-s − 0.583·47-s + 9/7·49-s + 0.274·53-s − 0.269·55-s + 0.781·59-s − 0.256·61-s − 0.124·65-s + 0.488·67-s − 0.712·71-s − 0.702·73-s − 0.911·77-s − 1.35·79-s − 1.75·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−15.19430159118139, −14.60679385796855, −14.13494608249605, −13.36558057967056, −13.02960788704147, −12.67954253008076, −11.92733766946608, −11.48783878091835, −11.01124076989610, −10.32848694263724, −9.664176735578647, −9.370602756792461, −8.810921489834340, −8.252796609363211, −7.278462669097675, −7.069893842534175, −6.549961696214957, −5.781591363579978, −5.342918030684769, −4.431241502873795, −3.861174095272976, −3.110358156213174, −2.969996078618612, −1.731617487289648, −0.8902393932025061, 0,
0.8902393932025061, 1.731617487289648, 2.969996078618612, 3.110358156213174, 3.861174095272976, 4.431241502873795, 5.342918030684769, 5.781591363579978, 6.549961696214957, 7.069893842534175, 7.278462669097675, 8.252796609363211, 8.810921489834340, 9.370602756792461, 9.664176735578647, 10.32848694263724, 11.01124076989610, 11.48783878091835, 11.92733766946608, 12.67954253008076, 13.02960788704147, 13.36558057967056, 14.13494608249605, 14.60679385796855, 15.19430159118139
