| L(s) = 1 | − 3-s + 9-s + 4·11-s − 2·13-s − 2·17-s − 4·19-s − 27-s + 2·29-s − 4·33-s − 10·37-s + 2·39-s + 10·41-s + 4·43-s − 8·47-s − 7·49-s + 2·51-s − 10·53-s + 4·57-s + 4·59-s + 2·61-s + 12·67-s − 8·71-s − 10·73-s + 81-s + 12·83-s − 2·87-s − 6·89-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.485·17-s − 0.917·19-s − 0.192·27-s + 0.371·29-s − 0.696·33-s − 1.64·37-s + 0.320·39-s + 1.56·41-s + 0.609·43-s − 1.16·47-s − 49-s + 0.280·51-s − 1.37·53-s + 0.529·57-s + 0.520·59-s + 0.256·61-s + 1.46·67-s − 0.949·71-s − 1.17·73-s + 1/9·81-s + 1.31·83-s − 0.214·87-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−7.907376134915276025352994775254, −6.95918819134071666698691519322, −6.54765356303077776143600578890, −5.83385105756922364211493953838, −4.87749252069353952041585154019, −4.29882241238245294759942818888, −3.45228302391592708746668800360, −2.28040616696634397835589842452, −1.33058973657865505623944366225, 0,
1.33058973657865505623944366225, 2.28040616696634397835589842452, 3.45228302391592708746668800360, 4.29882241238245294759942818888, 4.87749252069353952041585154019, 5.83385105756922364211493953838, 6.54765356303077776143600578890, 6.95918819134071666698691519322, 7.907376134915276025352994775254
