| L(s) = 1 | + 3-s + 9-s − 4·11-s + 2·13-s − 2·17-s + 2·19-s − 6·23-s + 27-s − 8·29-s + 8·31-s − 4·33-s + 6·37-s + 2·39-s + 2·41-s + 4·43-s + 6·47-s − 7·49-s − 2·51-s − 10·53-s + 2·57-s + 8·59-s + 2·61-s − 4·67-s − 6·69-s − 4·73-s + 4·79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 0.485·17-s + 0.458·19-s − 1.25·23-s + 0.192·27-s − 1.48·29-s + 1.43·31-s − 0.696·33-s + 0.986·37-s + 0.320·39-s + 0.312·41-s + 0.609·43-s + 0.875·47-s − 49-s − 0.280·51-s − 1.37·53-s + 0.264·57-s + 1.04·59-s + 0.256·61-s − 0.488·67-s − 0.722·69-s − 0.468·73-s + 0.450·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−7.65537979631675322478477404661, −6.68245442844013095171789908559, −5.96254332748508633041861889404, −5.33954684101116163788297772541, −4.42352469032320514140084794508, −3.85447286850495844935753266948, −2.88569338468783603911668354510, −2.34480792844601015085685424579, −1.33079787674644291615784960301, 0,
1.33079787674644291615784960301, 2.34480792844601015085685424579, 2.88569338468783603911668354510, 3.85447286850495844935753266948, 4.42352469032320514140084794508, 5.33954684101116163788297772541, 5.96254332748508633041861889404, 6.68245442844013095171789908559, 7.65537979631675322478477404661
