| L(s) = 1 | − 2·3-s − 2·5-s + 7-s + 9-s − 4·11-s − 6·13-s + 4·15-s − 4·17-s − 4·19-s − 2·21-s − 25-s + 4·27-s − 10·29-s − 2·31-s + 8·33-s − 2·35-s − 37-s + 12·39-s − 2·41-s + 4·43-s − 2·45-s + 49-s + 8·51-s + 6·53-s + 8·55-s + 8·57-s − 14·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 1.66·13-s + 1.03·15-s − 0.970·17-s − 0.917·19-s − 0.436·21-s − 1/5·25-s + 0.769·27-s − 1.85·29-s − 0.359·31-s + 1.39·33-s − 0.338·35-s − 0.164·37-s + 1.92·39-s − 0.312·41-s + 0.609·43-s − 0.298·45-s + 1/7·49-s + 1.12·51-s + 0.824·53-s + 1.07·55-s + 1.05·57-s − 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 37 | \( 1 + T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
| 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
−7.40292247585398890371813166493, −7.23149601414821600235262873758, −6.04588321780968239786873666056, −5.42174750212442718983997255179, −4.70284925629556994475140295712, −4.17565602178245282151914638543, −2.86110434877472720495424861629, −1.96416416772468748823915300367, 0, 0,
1.96416416772468748823915300367, 2.86110434877472720495424861629, 4.17565602178245282151914638543, 4.70284925629556994475140295712, 5.42174750212442718983997255179, 6.04588321780968239786873666056, 7.23149601414821600235262873758, 7.40292247585398890371813166493
