| L(s) = 1 | + 4·5-s + 4·7-s + 4·11-s + 2·13-s − 6·17-s + 2·19-s + 8·23-s + 11·25-s − 6·29-s − 8·31-s + 16·35-s + 2·37-s + 6·41-s − 2·43-s + 12·47-s + 9·49-s + 4·53-s + 16·55-s + 12·59-s − 2·61-s + 8·65-s + 10·67-s + 8·71-s + 14·73-s + 16·77-s + 6·79-s − 83-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 1.51·7-s + 1.20·11-s + 0.554·13-s − 1.45·17-s + 0.458·19-s + 1.66·23-s + 11/5·25-s − 1.11·29-s − 1.43·31-s + 2.70·35-s + 0.328·37-s + 0.937·41-s − 0.304·43-s + 1.75·47-s + 9/7·49-s + 0.549·53-s + 2.15·55-s + 1.56·59-s − 0.256·61-s + 0.992·65-s + 1.22·67-s + 0.949·71-s + 1.63·73-s + 1.82·77-s + 0.675·79-s − 0.109·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.132034808\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.132034808\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 83 | \( 1 + T \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 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
−14.40021161631272, −14.13320917387039, −13.62332390815200, −13.09811405281652, −12.75293881765776, −11.92450400962766, −11.20279428051853, −10.98132736586849, −10.70186236300950, −9.636358940106619, −9.329238097622300, −8.900075955919158, −8.530015736241336, −7.624473643664474, −6.935873208801501, −6.638969898879048, −5.860148820984525, −5.325792032981029, −5.032864777518640, −4.151636205421127, −3.658411766788967, −2.382410633103923, −2.217957880230247, −1.350114848241906, −1.006088138735066,
1.006088138735066, 1.350114848241906, 2.217957880230247, 2.382410633103923, 3.658411766788967, 4.151636205421127, 5.032864777518640, 5.325792032981029, 5.860148820984525, 6.638969898879048, 6.935873208801501, 7.624473643664474, 8.530015736241336, 8.900075955919158, 9.329238097622300, 9.636358940106619, 10.70186236300950, 10.98132736586849, 11.20279428051853, 11.92450400962766, 12.75293881765776, 13.09811405281652, 13.62332390815200, 14.13320917387039, 14.40021161631272
