| L(s) = 1 | − 4·5-s − 3·7-s − 5·11-s − 3·13-s − 3·17-s − 7·19-s + 9·23-s + 11·25-s + 2·31-s + 12·35-s − 37-s − 6·41-s + 4·43-s − 10·47-s + 2·49-s + 3·53-s + 20·55-s + 4·59-s + 2·61-s + 12·65-s + 6·67-s − 12·71-s + 13·73-s + 15·77-s + 6·79-s − 5·83-s + 12·85-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 1.13·7-s − 1.50·11-s − 0.832·13-s − 0.727·17-s − 1.60·19-s + 1.87·23-s + 11/5·25-s + 0.359·31-s + 2.02·35-s − 0.164·37-s − 0.937·41-s + 0.609·43-s − 1.45·47-s + 2/7·49-s + 0.412·53-s + 2.69·55-s + 0.520·59-s + 0.256·61-s + 1.48·65-s + 0.733·67-s − 1.42·71-s + 1.52·73-s + 1.70·77-s + 0.675·79-s − 0.548·83-s + 1.30·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 37 | \( 1 + T \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 13 T + p T^{2} \) | 1.73.an |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + 5 T + p T^{2} \) | 1.83.f |
| 89 | \( 1 + 11 T + p T^{2} \) | 1.89.l |
| 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
−15.71005075469626, −15.34764916444077, −15.05233431693345, −14.45390894149339, −13.37592818457475, −12.96540567445059, −12.68899115352405, −12.16533927202433, −11.29579688361062, −11.06528593687300, −10.36437030485847, −9.914480881525380, −8.972025424038701, −8.553514663911738, −7.988152965673084, −7.352571715916406, −6.857272169997633, −6.415572561465275, −5.274530747401378, −4.773071341311620, −4.215269188156720, −3.357818236688999, −2.928270372014711, −2.210318530797892, −0.5826217374875523, 0,
0.5826217374875523, 2.210318530797892, 2.928270372014711, 3.357818236688999, 4.215269188156720, 4.773071341311620, 5.274530747401378, 6.415572561465275, 6.857272169997633, 7.352571715916406, 7.988152965673084, 8.553514663911738, 8.972025424038701, 9.914480881525380, 10.36437030485847, 11.06528593687300, 11.29579688361062, 12.16533927202433, 12.68899115352405, 12.96540567445059, 13.37592818457475, 14.45390894149339, 15.05233431693345, 15.34764916444077, 15.71005075469626
