| L(s) = 1 | + 3-s − 2·7-s + 9-s − 13-s − 8·17-s + 8·19-s − 2·21-s + 27-s + 6·29-s + 8·31-s + 10·37-s − 39-s − 2·41-s + 4·43-s − 6·47-s − 3·49-s − 8·51-s − 6·53-s + 8·57-s + 4·59-s − 6·61-s − 2·63-s − 12·67-s + 2·73-s − 8·79-s + 81-s − 8·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.277·13-s − 1.94·17-s + 1.83·19-s − 0.436·21-s + 0.192·27-s + 1.11·29-s + 1.43·31-s + 1.64·37-s − 0.160·39-s − 0.312·41-s + 0.609·43-s − 0.875·47-s − 3/7·49-s − 1.12·51-s − 0.824·53-s + 1.05·57-s + 0.520·59-s − 0.768·61-s − 0.251·63-s − 1.46·67-s + 0.234·73-s − 0.900·79-s + 1/9·81-s − 0.878·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 8 T + p T^{2} \) | 1.17.i |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−13.78852965168229, −13.28992777019752, −12.95895446668061, −12.41369144286950, −11.72710755250818, −11.51000202896859, −10.86513475447335, −10.20800513707660, −9.779127875912548, −9.473458508626198, −8.865753963281276, −8.496633670843624, −7.768309758426553, −7.458622459676099, −6.756216461866498, −6.349578945267080, −5.964683618337198, −4.926763428795480, −4.706693339328801, −4.083353399161028, −3.352158723715251, −2.747721703488105, −2.575892505060364, −1.571241897748496, −0.8938174134034615, 0,
0.8938174134034615, 1.571241897748496, 2.575892505060364, 2.747721703488105, 3.352158723715251, 4.083353399161028, 4.706693339328801, 4.926763428795480, 5.964683618337198, 6.349578945267080, 6.756216461866498, 7.458622459676099, 7.768309758426553, 8.496633670843624, 8.865753963281276, 9.473458508626198, 9.779127875912548, 10.20800513707660, 10.86513475447335, 11.51000202896859, 11.72710755250818, 12.41369144286950, 12.95895446668061, 13.28992777019752, 13.78852965168229
