| L(s) = 1 | − 3·3-s + 4·5-s − 5·7-s + 6·9-s − 3·11-s − 4·13-s − 12·15-s + 4·19-s + 15·21-s + 11·25-s − 9·27-s − 5·29-s − 5·31-s + 9·33-s − 20·35-s + 3·37-s + 12·39-s + 2·41-s + 4·43-s + 24·45-s + 2·47-s + 18·49-s − 6·53-s − 12·55-s − 12·57-s − 7·59-s + 15·61-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1.78·5-s − 1.88·7-s + 2·9-s − 0.904·11-s − 1.10·13-s − 3.09·15-s + 0.917·19-s + 3.27·21-s + 11/5·25-s − 1.73·27-s − 0.928·29-s − 0.898·31-s + 1.56·33-s − 3.38·35-s + 0.493·37-s + 1.92·39-s + 0.312·41-s + 0.609·43-s + 3.57·45-s + 0.291·47-s + 18/7·49-s − 0.824·53-s − 1.61·55-s − 1.58·57-s − 0.911·59-s + 1.92·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 383792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 383792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6938446682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6938446682\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 17 | \( 1 \) | |
| 83 | \( 1 - T \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 + 5 T + p T^{2} \) | 1.7.f |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 7 T + p T^{2} \) | 1.59.h |
| 61 | \( 1 - 15 T + p T^{2} \) | 1.61.ap |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−12.44458055731872, −12.33416948314123, −11.43779843111906, −11.09786514754169, −10.54654101516035, −10.18428289160647, −9.784190668058005, −9.544933167202070, −9.298460006926699, −8.502892039089171, −7.475690548581485, −7.249037699786702, −6.836897892813218, −6.242923716231259, −5.948348979803783, −5.592445805477504, −5.215249696746548, −4.833027783627557, −4.036987594293575, −3.377430782487296, −2.678648394498923, −2.382581084825469, −1.630523924676149, −0.8869118785324670, −0.2880143603433454,
0.2880143603433454, 0.8869118785324670, 1.630523924676149, 2.382581084825469, 2.678648394498923, 3.377430782487296, 4.036987594293575, 4.833027783627557, 5.215249696746548, 5.592445805477504, 5.948348979803783, 6.242923716231259, 6.836897892813218, 7.249037699786702, 7.475690548581485, 8.502892039089171, 9.298460006926699, 9.544933167202070, 9.784190668058005, 10.18428289160647, 10.54654101516035, 11.09786514754169, 11.43779843111906, 12.33416948314123, 12.44458055731872
