| L(s) = 1 | − 3-s + 7-s + 9-s + 11-s − 13-s − 17-s + 4·19-s − 21-s + 3·23-s − 27-s − 8·29-s + 4·31-s − 33-s + 3·37-s + 39-s − 9·41-s + 8·43-s − 10·47-s − 6·49-s + 51-s − 53-s − 4·57-s − 4·59-s − 11·61-s + 63-s + 4·67-s − 3·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.277·13-s − 0.242·17-s + 0.917·19-s − 0.218·21-s + 0.625·23-s − 0.192·27-s − 1.48·29-s + 0.718·31-s − 0.174·33-s + 0.493·37-s + 0.160·39-s − 1.40·41-s + 1.21·43-s − 1.45·47-s − 6/7·49-s + 0.140·51-s − 0.137·53-s − 0.529·57-s − 0.520·59-s − 1.40·61-s + 0.125·63-s + 0.488·67-s − 0.361·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 + 15 T + p T^{2} \) | 1.97.p |
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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
−16.40252224430618, −15.68393786596513, −15.20165346494817, −14.71828423355684, −13.96428625769979, −13.59576253590032, −12.74470960777585, −12.47282930158314, −11.55521808342728, −11.36679953648165, −10.81104929007508, −9.957866394747847, −9.561132628105005, −8.939272179261127, −8.120507365711653, −7.596962963825611, −6.938472768293286, −6.359362864666505, −5.619325146011444, −5.026591275762476, −4.488271251630112, −3.626583576680202, −2.902278660026820, −1.863886817778339, −1.144358120405309, 0,
1.144358120405309, 1.863886817778339, 2.902278660026820, 3.626583576680202, 4.488271251630112, 5.026591275762476, 5.619325146011444, 6.359362864666505, 6.938472768293286, 7.596962963825611, 8.120507365711653, 8.939272179261127, 9.561132628105005, 9.957866394747847, 10.81104929007508, 11.36679953648165, 11.55521808342728, 12.47282930158314, 12.74470960777585, 13.59576253590032, 13.96428625769979, 14.71828423355684, 15.20165346494817, 15.68393786596513, 16.40252224430618
