| L(s) = 1 | − 2·5-s + 2·7-s − 4·11-s + 2·13-s + 2·17-s + 2·19-s − 23-s − 25-s − 4·29-s − 4·35-s − 2·37-s + 2·43-s + 12·47-s − 3·49-s + 2·53-s + 8·55-s − 12·59-s + 14·61-s − 4·65-s − 2·67-s + 6·73-s − 8·77-s + 6·79-s − 4·83-s − 4·85-s − 18·89-s + 4·91-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.755·7-s − 1.20·11-s + 0.554·13-s + 0.485·17-s + 0.458·19-s − 0.208·23-s − 1/5·25-s − 0.742·29-s − 0.676·35-s − 0.328·37-s + 0.304·43-s + 1.75·47-s − 3/7·49-s + 0.274·53-s + 1.07·55-s − 1.56·59-s + 1.79·61-s − 0.496·65-s − 0.244·67-s + 0.702·73-s − 0.911·77-s + 0.675·79-s − 0.439·83-s − 0.433·85-s − 1.90·89-s + 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 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.ac |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−16.39907967045005, −15.83819853766549, −15.43765151548754, −15.01496150310549, −14.15578558308561, −13.85219243522250, −13.06185501252217, −12.54573779518517, −11.89176432902059, −11.39457711541294, −10.87141289150156, −10.35410115469433, −9.625481870497654, −8.842812211696589, −8.237074899973337, −7.671439480571715, −7.460034566184015, −6.473203821116557, −5.531651524982126, −5.249027388929914, −4.298462460722993, −3.768390663809094, −2.954739428627474, −2.099468896007230, −1.095271496911977, 0,
1.095271496911977, 2.099468896007230, 2.954739428627474, 3.768390663809094, 4.298462460722993, 5.249027388929914, 5.531651524982126, 6.473203821116557, 7.460034566184015, 7.671439480571715, 8.237074899973337, 8.842812211696589, 9.625481870497654, 10.35410115469433, 10.87141289150156, 11.39457711541294, 11.89176432902059, 12.54573779518517, 13.06185501252217, 13.85219243522250, 14.15578558308561, 15.01496150310549, 15.43765151548754, 15.83819853766549, 16.39907967045005
