| L(s) = 1 | + 5-s − 2·11-s − 2·13-s + 6·17-s + 2·19-s + 8·23-s + 25-s − 4·29-s − 2·31-s − 2·37-s − 6·41-s + 4·43-s − 4·47-s − 2·53-s − 2·55-s − 4·59-s + 12·61-s − 2·65-s − 16·67-s + 6·71-s − 14·73-s − 8·79-s + 4·83-s + 6·85-s − 10·89-s + 2·95-s − 10·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.603·11-s − 0.554·13-s + 1.45·17-s + 0.458·19-s + 1.66·23-s + 1/5·25-s − 0.742·29-s − 0.359·31-s − 0.328·37-s − 0.937·41-s + 0.609·43-s − 0.583·47-s − 0.274·53-s − 0.269·55-s − 0.520·59-s + 1.53·61-s − 0.248·65-s − 1.95·67-s + 0.712·71-s − 1.63·73-s − 0.900·79-s + 0.439·83-s + 0.650·85-s − 1.05·89-s + 0.205·95-s − 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| good | 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 + 16 T + p T^{2} \) | 1.67.q |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−15.02042014149651, −14.70957537244685, −14.25124006411624, −13.51801431500297, −13.16713747608914, −12.59710596187512, −12.13271292141869, −11.49511550403960, −10.94659827267122, −10.34753921448638, −9.902755539357785, −9.388665928132748, −8.834046084223465, −8.169455934699961, −7.486355681930116, −7.193142520170203, −6.476904705087811, −5.568903416282852, −5.388347549407184, −4.781292652253008, −3.906310636817538, −3.068360734741656, −2.793480976652291, −1.734572658947701, −1.114188796941951, 0,
1.114188796941951, 1.734572658947701, 2.793480976652291, 3.068360734741656, 3.906310636817538, 4.781292652253008, 5.388347549407184, 5.568903416282852, 6.476904705087811, 7.193142520170203, 7.486355681930116, 8.169455934699961, 8.834046084223465, 9.388665928132748, 9.902755539357785, 10.34753921448638, 10.94659827267122, 11.49511550403960, 12.13271292141869, 12.59710596187512, 13.16713747608914, 13.51801431500297, 14.25124006411624, 14.70957537244685, 15.02042014149651
