| L(s) = 1 | − 3-s − 4·7-s + 9-s + 4·11-s − 2·13-s + 2·17-s + 4·19-s + 4·21-s − 8·23-s − 27-s − 2·29-s − 4·33-s + 37-s + 2·39-s + 10·41-s − 4·43-s + 9·49-s − 2·51-s − 2·53-s − 4·57-s − 4·59-s + 14·61-s − 4·63-s + 4·67-s + 8·69-s − 8·71-s − 2·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 0.485·17-s + 0.917·19-s + 0.872·21-s − 1.66·23-s − 0.192·27-s − 0.371·29-s − 0.696·33-s + 0.164·37-s + 0.320·39-s + 1.56·41-s − 0.609·43-s + 9/7·49-s − 0.280·51-s − 0.274·53-s − 0.529·57-s − 0.520·59-s + 1.79·61-s − 0.503·63-s + 0.488·67-s + 0.963·69-s − 0.949·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 37 | \( 1 - T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 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.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.29876927928288, −12.79407001342461, −12.49199528692446, −12.00441275218934, −11.53910575281136, −11.31008981774520, −10.31005509924655, −10.09252021979922, −9.696003949932298, −9.277866799004316, −8.820780579503980, −8.045410372800154, −7.459447754294429, −7.127388332238451, −6.508068948234552, −6.044787646992258, −5.839043310069122, −5.106999942874849, −4.453750522198070, −3.785112032074320, −3.590183337649622, −2.827423636281752, −2.191382274198890, −1.390953518392424, −0.7105093902115223, 0,
0.7105093902115223, 1.390953518392424, 2.191382274198890, 2.827423636281752, 3.590183337649622, 3.785112032074320, 4.453750522198070, 5.106999942874849, 5.839043310069122, 6.044787646992258, 6.508068948234552, 7.127388332238451, 7.459447754294429, 8.045410372800154, 8.820780579503980, 9.277866799004316, 9.696003949932298, 10.09252021979922, 10.31005509924655, 11.31008981774520, 11.53910575281136, 12.00441275218934, 12.49199528692446, 12.79407001342461, 13.29876927928288
