| L(s) = 1 | + 2·3-s + 9-s − 4·11-s + 4·13-s + 2·17-s + 6·19-s + 8·23-s − 5·25-s − 4·27-s + 2·29-s + 4·31-s − 8·33-s + 10·37-s + 8·39-s + 10·41-s + 4·43-s − 4·47-s + 4·51-s − 2·53-s + 12·57-s − 10·59-s + 8·61-s − 8·67-s + 16·69-s + 6·73-s − 10·75-s − 16·79-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s − 1.20·11-s + 1.10·13-s + 0.485·17-s + 1.37·19-s + 1.66·23-s − 25-s − 0.769·27-s + 0.371·29-s + 0.718·31-s − 1.39·33-s + 1.64·37-s + 1.28·39-s + 1.56·41-s + 0.609·43-s − 0.583·47-s + 0.560·51-s − 0.274·53-s + 1.58·57-s − 1.30·59-s + 1.02·61-s − 0.977·67-s + 1.92·69-s + 0.702·73-s − 1.15·75-s − 1.80·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.596644279\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.596644279\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−9.373857617588295367961483073830, −8.589272005710336492190097424760, −7.84325882583011970892071619935, −7.43617998805416894494930631088, −6.10276118363026386768152091852, −5.35174774645365978518414756780, −4.21868092620292662247305328011, −3.12022467334904801892948222253, −2.67787006845321594312840497948, −1.16036401648900510175062615275,
1.16036401648900510175062615275, 2.67787006845321594312840497948, 3.12022467334904801892948222253, 4.21868092620292662247305328011, 5.35174774645365978518414756780, 6.10276118363026386768152091852, 7.43617998805416894494930631088, 7.84325882583011970892071619935, 8.589272005710336492190097424760, 9.373857617588295367961483073830
