| L(s) = 1 | − 3-s + 5-s + 9-s − 2·11-s − 13-s − 15-s + 3·19-s + 25-s − 27-s + 9·31-s + 2·33-s − 3·37-s + 39-s + 2·41-s + 3·43-s + 45-s + 6·47-s − 2·55-s − 3·57-s − 4·59-s − 2·61-s − 65-s + 5·67-s − 14·71-s + 73-s − 75-s − 9·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.603·11-s − 0.277·13-s − 0.258·15-s + 0.688·19-s + 1/5·25-s − 0.192·27-s + 1.61·31-s + 0.348·33-s − 0.493·37-s + 0.160·39-s + 0.312·41-s + 0.457·43-s + 0.149·45-s + 0.875·47-s − 0.269·55-s − 0.397·57-s − 0.520·59-s − 0.256·61-s − 0.124·65-s + 0.610·67-s − 1.66·71-s + 0.117·73-s − 0.115·75-s − 1.01·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| good | 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 9 T + p T^{2} \) | 1.31.aj |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 3 T + p T^{2} \) | 1.43.ad |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 + 9 T + p T^{2} \) | 1.79.j |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
| 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
−14.84205940220380, −14.27008768437983, −13.74202074656603, −13.36534440390961, −12.76775847194166, −12.19551703042256, −11.86791406026061, −11.19945715124421, −10.67153893192126, −10.18507353133971, −9.773141421397859, −9.169464646537553, −8.566003757330916, −7.880782991313351, −7.427428584512666, −6.763862407871538, −6.274825763127747, −5.520926689823747, −5.312376207729162, −4.492954398335809, −4.038074275829154, −2.939114939178374, −2.653399718705438, −1.648801289769559, −0.9668000273640634, 0,
0.9668000273640634, 1.648801289769559, 2.653399718705438, 2.939114939178374, 4.038074275829154, 4.492954398335809, 5.312376207729162, 5.520926689823747, 6.274825763127747, 6.763862407871538, 7.427428584512666, 7.880782991313351, 8.566003757330916, 9.169464646537553, 9.773141421397859, 10.18507353133971, 10.67153893192126, 11.19945715124421, 11.86791406026061, 12.19551703042256, 12.76775847194166, 13.36534440390961, 13.74202074656603, 14.27008768437983, 14.84205940220380
