| L(s) = 1 | + 3-s + 4·5-s + 2·7-s + 9-s − 4·11-s + 4·15-s + 6·17-s − 19-s + 2·21-s − 8·23-s + 11·25-s + 27-s + 6·29-s + 2·31-s − 4·33-s + 8·35-s − 10·37-s + 2·41-s + 43-s + 4·45-s − 4·47-s − 3·49-s + 6·51-s + 12·53-s − 16·55-s − 57-s − 8·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.78·5-s + 0.755·7-s + 1/3·9-s − 1.20·11-s + 1.03·15-s + 1.45·17-s − 0.229·19-s + 0.436·21-s − 1.66·23-s + 11/5·25-s + 0.192·27-s + 1.11·29-s + 0.359·31-s − 0.696·33-s + 1.35·35-s − 1.64·37-s + 0.312·41-s + 0.152·43-s + 0.596·45-s − 0.583·47-s − 3/7·49-s + 0.840·51-s + 1.64·53-s − 2.15·55-s − 0.132·57-s − 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.081184529\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.081184529\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 19 | \( 1 + T \) | |
| 43 | \( 1 - T \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.56466528051438, −14.21833492538485, −13.78289086469956, −13.50539892630915, −12.83236695024835, −12.24984233681446, −11.88589754170510, −10.85315381932149, −10.36112459910417, −10.07932639419062, −9.731233474344369, −8.838918646580301, −8.512178707301047, −7.818208544710057, −7.471560159642885, −6.526007191533794, −6.032358550283470, −5.432054281392185, −5.039783605804137, −4.368909045043706, −3.340071183570091, −2.814095324042963, −1.998794965076084, −1.780399746126638, −0.7866628839346186,
0.7866628839346186, 1.780399746126638, 1.998794965076084, 2.814095324042963, 3.340071183570091, 4.368909045043706, 5.039783605804137, 5.432054281392185, 6.032358550283470, 6.526007191533794, 7.471560159642885, 7.818208544710057, 8.512178707301047, 8.838918646580301, 9.731233474344369, 10.07932639419062, 10.36112459910417, 10.85315381932149, 11.88589754170510, 12.24984233681446, 12.83236695024835, 13.50539892630915, 13.78289086469956, 14.21833492538485, 14.56466528051438
