| L(s) = 1 | + 2·11-s + 6·13-s − 6·17-s + 6·19-s − 23-s − 5·25-s − 2·29-s − 8·31-s + 8·37-s + 6·41-s + 10·43-s − 8·47-s + 4·53-s + 8·59-s + 14·67-s + 2·73-s − 8·79-s + 14·83-s − 10·89-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.603·11-s + 1.66·13-s − 1.45·17-s + 1.37·19-s − 0.208·23-s − 25-s − 0.371·29-s − 1.43·31-s + 1.31·37-s + 0.937·41-s + 1.52·43-s − 1.16·47-s + 0.549·53-s + 1.04·59-s + 1.71·67-s + 0.234·73-s − 0.900·79-s + 1.53·83-s − 1.05·89-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.832846625\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.832846625\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 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
−13.99547648784672, −13.43941440595845, −13.03511394972825, −12.67875040421894, −11.77818346060426, −11.45297055344005, −11.07137102670688, −10.70686792950294, −9.795905079698140, −9.405547895073235, −9.037183398016819, −8.450058922917410, −7.883430606246740, −7.368774377747053, −6.745470502329921, −6.216649966327376, −5.750774899992629, −5.253804620080238, −4.333633813150680, −3.902142372076536, −3.543507036930398, −2.625041999305137, −1.984869247334284, −1.269667250853434, −0.5851987337539213,
0.5851987337539213, 1.269667250853434, 1.984869247334284, 2.625041999305137, 3.543507036930398, 3.902142372076536, 4.333633813150680, 5.253804620080238, 5.750774899992629, 6.216649966327376, 6.745470502329921, 7.368774377747053, 7.883430606246740, 8.450058922917410, 9.037183398016819, 9.405547895073235, 9.795905079698140, 10.70686792950294, 11.07137102670688, 11.45297055344005, 11.77818346060426, 12.67875040421894, 13.03511394972825, 13.43941440595845, 13.99547648784672
