| L(s) = 1 | − 3-s + 4·7-s + 9-s + 11-s − 6·13-s − 2·17-s − 4·19-s − 4·21-s − 4·23-s − 27-s − 6·29-s − 33-s + 6·37-s + 6·39-s − 6·41-s + 4·43-s + 12·47-s + 9·49-s + 2·51-s + 2·53-s + 4·57-s − 12·59-s + 14·61-s + 4·63-s + 4·67-s + 4·69-s − 12·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.51·7-s + 1/3·9-s + 0.301·11-s − 1.66·13-s − 0.485·17-s − 0.917·19-s − 0.872·21-s − 0.834·23-s − 0.192·27-s − 1.11·29-s − 0.174·33-s + 0.986·37-s + 0.960·39-s − 0.937·41-s + 0.609·43-s + 1.75·47-s + 9/7·49-s + 0.280·51-s + 0.274·53-s + 0.529·57-s − 1.56·59-s + 1.79·61-s + 0.503·63-s + 0.488·67-s + 0.481·69-s − 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.394175526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.394175526\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 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 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−14.46552023717534, −14.15665379027187, −13.39013369919956, −12.88024619983472, −12.21766264301747, −11.92165092084954, −11.43492073431649, −10.87869936949595, −10.49916654201778, −9.853147755650887, −9.306168961860411, −8.695874481518807, −8.123680690193615, −7.460453812151813, −7.294847054233136, −6.433697752250209, −5.855724923172602, −5.249737578708856, −4.726841129785682, −4.305282628073441, −3.724632757646446, −2.425932266800769, −2.187144852921480, −1.409790225606539, −0.4241025706230641,
0.4241025706230641, 1.409790225606539, 2.187144852921480, 2.425932266800769, 3.724632757646446, 4.305282628073441, 4.726841129785682, 5.249737578708856, 5.855724923172602, 6.433697752250209, 7.294847054233136, 7.460453812151813, 8.123680690193615, 8.695874481518807, 9.306168961860411, 9.853147755650887, 10.49916654201778, 10.87869936949595, 11.43492073431649, 11.92165092084954, 12.21766264301747, 12.88024619983472, 13.39013369919956, 14.15665379027187, 14.46552023717534
