| L(s) = 1 | + 3-s + 7-s + 9-s − 13-s + 4·19-s + 21-s − 6·23-s + 27-s − 6·29-s + 4·31-s − 2·37-s − 39-s − 6·41-s + 8·43-s + 6·47-s + 49-s + 6·53-s + 4·57-s + 6·59-s − 10·61-s + 63-s − 4·67-s − 6·69-s + 12·71-s + 10·73-s − 8·79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.277·13-s + 0.917·19-s + 0.218·21-s − 1.25·23-s + 0.192·27-s − 1.11·29-s + 0.718·31-s − 0.328·37-s − 0.160·39-s − 0.937·41-s + 1.21·43-s + 0.875·47-s + 1/7·49-s + 0.824·53-s + 0.529·57-s + 0.781·59-s − 1.28·61-s + 0.125·63-s − 0.488·67-s − 0.722·69-s + 1.42·71-s + 1.17·73-s − 0.900·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 109200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 109200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.102197989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.102197989\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.82187457188943, −13.33860004709540, −12.65429121190057, −12.15201898080073, −11.85617086439206, −11.24393602891880, −10.67269809120763, −10.20324457733566, −9.669315001034185, −9.266985719979514, −8.721349107197045, −8.156718782482599, −7.697957184382760, −7.338211397113307, −6.706892271560900, −6.067557113915590, −5.466400481893954, −5.045141154005686, −4.257104618412017, −3.865121205926997, −3.252502949839358, −2.518392465091416, −2.045932770024718, −1.341959442205734, −0.5294553902532334,
0.5294553902532334, 1.341959442205734, 2.045932770024718, 2.518392465091416, 3.252502949839358, 3.865121205926997, 4.257104618412017, 5.045141154005686, 5.466400481893954, 6.067557113915590, 6.706892271560900, 7.338211397113307, 7.697957184382760, 8.156718782482599, 8.721349107197045, 9.266985719979514, 9.669315001034185, 10.20324457733566, 10.67269809120763, 11.24393602891880, 11.85617086439206, 12.15201898080073, 12.65429121190057, 13.33860004709540, 13.82187457188943
