| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s + 5·11-s + 4·13-s + 15-s + 17-s + 19-s − 21-s + 8·23-s + 25-s − 27-s − 29-s − 5·33-s − 35-s − 37-s − 4·39-s − 5·41-s + 4·43-s − 45-s − 3·47-s − 6·49-s − 51-s − 5·53-s − 5·55-s − 57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.50·11-s + 1.10·13-s + 0.258·15-s + 0.242·17-s + 0.229·19-s − 0.218·21-s + 1.66·23-s + 1/5·25-s − 0.192·27-s − 0.185·29-s − 0.870·33-s − 0.169·35-s − 0.164·37-s − 0.640·39-s − 0.780·41-s + 0.609·43-s − 0.149·45-s − 0.437·47-s − 6/7·49-s − 0.140·51-s − 0.686·53-s − 0.674·55-s − 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.886424026\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.886424026\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 17 | \( 1 - T \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 5 T + p T^{2} \) | 1.53.f |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + 6 T + p T^{2} \) | 1.67.g |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−8.497259520552686283758893592707, −7.65081306740013705831673968649, −6.80552898931082038790525195431, −6.37687863619112417808931811511, −5.43232663137048254531984962980, −4.68030847881529981322695477505, −3.85400346631121934257234186846, −3.19632012454780569948363796912, −1.62807602814767249117011534968, −0.894831587803758256927277281709,
0.894831587803758256927277281709, 1.62807602814767249117011534968, 3.19632012454780569948363796912, 3.85400346631121934257234186846, 4.68030847881529981322695477505, 5.43232663137048254531984962980, 6.37687863619112417808931811511, 6.80552898931082038790525195431, 7.65081306740013705831673968649, 8.497259520552686283758893592707
