| L(s) = 1 | + 3-s − 2·5-s − 4·7-s + 9-s − 2·13-s − 2·15-s − 17-s + 4·19-s − 4·21-s − 4·23-s − 25-s + 27-s − 6·29-s − 4·31-s + 8·35-s − 2·37-s − 2·39-s − 10·41-s + 4·43-s − 2·45-s − 8·47-s + 9·49-s − 51-s − 6·53-s + 4·57-s + 4·59-s − 6·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 1.51·7-s + 1/3·9-s − 0.554·13-s − 0.516·15-s − 0.242·17-s + 0.917·19-s − 0.872·21-s − 0.834·23-s − 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.718·31-s + 1.35·35-s − 0.328·37-s − 0.320·39-s − 1.56·41-s + 0.609·43-s − 0.298·45-s − 1.16·47-s + 9/7·49-s − 0.140·51-s − 0.824·53-s + 0.529·57-s + 0.520·59-s − 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 394944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 394944 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−12.62305633682296, −12.26941167581946, −11.78413921389836, −11.49789034328657, −10.83661695433780, −10.25360255213651, −9.911973263140994, −9.534220658128367, −9.061458673733449, −8.692795313180805, −8.016434300504580, −7.599773337181440, −7.281540705507690, −6.838089048273408, −6.240671268435029, −5.806769185772610, −5.213243681132860, −4.588183433779361, −4.020803606361778, −3.629282181426829, −3.102181801798650, −2.915316326084700, −1.953116814601460, −1.583425720220975, −0.4473612773819170, 0,
0.4473612773819170, 1.583425720220975, 1.953116814601460, 2.915316326084700, 3.102181801798650, 3.629282181426829, 4.020803606361778, 4.588183433779361, 5.213243681132860, 5.806769185772610, 6.240671268435029, 6.838089048273408, 7.281540705507690, 7.599773337181440, 8.016434300504580, 8.692795313180805, 9.061458673733449, 9.534220658128367, 9.911973263140994, 10.25360255213651, 10.83661695433780, 11.49789034328657, 11.78413921389836, 12.26941167581946, 12.62305633682296
