| L(s) = 1 | − 2-s + 4-s + 2·7-s − 8-s − 2·14-s + 16-s − 2·19-s − 6·23-s + 2·28-s + 4·31-s − 32-s + 2·37-s + 2·38-s − 6·41-s + 4·43-s + 6·46-s − 3·49-s − 6·53-s − 2·56-s − 10·61-s − 4·62-s + 64-s + 8·67-s + 8·73-s − 2·74-s − 2·76-s + 8·79-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s − 0.534·14-s + 1/4·16-s − 0.458·19-s − 1.25·23-s + 0.377·28-s + 0.718·31-s − 0.176·32-s + 0.328·37-s + 0.324·38-s − 0.937·41-s + 0.609·43-s + 0.884·46-s − 3/7·49-s − 0.824·53-s − 0.267·56-s − 1.28·61-s − 0.508·62-s + 1/8·64-s + 0.977·67-s + 0.936·73-s − 0.232·74-s − 0.229·76-s + 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.444300838\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.444300838\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 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.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−14.15845088187538, −13.62159392401250, −13.03196092229249, −12.37581706817451, −11.98953653100809, −11.52032946366241, −10.93788165403497, −10.58938888897878, −9.972327482950616, −9.559578711934711, −8.935918764131013, −8.421708791648817, −7.889257630046265, −7.689886130190927, −6.873858338611803, −6.262353548440994, −5.980233415737615, −5.003014003770678, −4.731795872363052, −3.885049635782390, −3.344930241392822, −2.428850537208762, −1.988536096291695, −1.295074643959783, −0.4546659754875194,
0.4546659754875194, 1.295074643959783, 1.988536096291695, 2.428850537208762, 3.344930241392822, 3.885049635782390, 4.731795872363052, 5.003014003770678, 5.980233415737615, 6.262353548440994, 6.873858338611803, 7.689886130190927, 7.889257630046265, 8.421708791648817, 8.935918764131013, 9.559578711934711, 9.972327482950616, 10.58938888897878, 10.93788165403497, 11.52032946366241, 11.98953653100809, 12.37581706817451, 13.03196092229249, 13.62159392401250, 14.15845088187538
