| L(s) = 1 | + 475·3-s + 1.02e3·4-s − 3.00e3·5-s + 1.66e5·9-s − 1.61e5·11-s + 4.86e5·12-s − 1.42e6·15-s + 1.04e6·16-s − 3.07e6·20-s − 1.19e7·23-s − 7.59e5·25-s + 5.10e7·27-s + 3.19e6·31-s − 7.64e7·33-s + 1.70e8·36-s − 1.37e8·37-s − 1.64e8·44-s − 4.99e8·45-s + 1.51e8·47-s + 4.98e8·48-s + 2.82e8·49-s + 3.75e8·53-s + 4.83e8·55-s − 8.13e8·59-s − 1.45e9·60-s + 1.07e9·64-s + 2.61e9·67-s + ⋯ |
| L(s) = 1 | + 1.95·3-s + 4-s − 0.960·5-s + 2.82·9-s − 11-s + 1.95·12-s − 1.87·15-s + 16-s − 0.960·20-s − 1.85·23-s − 0.0777·25-s + 3.55·27-s + 0.111·31-s − 1.95·33-s + 2.82·36-s − 1.97·37-s − 44-s − 2.70·45-s + 0.661·47-s + 1.95·48-s + 49-s + 0.896·53-s + 0.960·55-s − 1.13·59-s − 1.87·60-s + 64-s + 1.93·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(3.006532895\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.006532895\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + p^{5} T \) |
| good | 2 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 3 | \( 1 - 475 T + p^{10} T^{2} \) |
| 5 | \( 1 + 3001 T + p^{10} T^{2} \) |
| 7 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 13 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 17 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 19 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 23 | \( 1 + 11910325 T + p^{10} T^{2} \) |
| 29 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 31 | \( 1 - 3192323 T + p^{10} T^{2} \) |
| 37 | \( 1 + 137082625 T + p^{10} T^{2} \) |
| 41 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 43 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 47 | \( 1 - 151795250 T + p^{10} T^{2} \) |
| 53 | \( 1 - 375066650 T + p^{10} T^{2} \) |
| 59 | \( 1 + 813567973 T + p^{10} T^{2} \) |
| 61 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 67 | \( 1 - 2616638675 T + p^{10} T^{2} \) |
| 71 | \( 1 - 783651827 T + p^{10} T^{2} \) |
| 73 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 79 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 83 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 89 | \( 1 + 2870912977 T + p^{10} T^{2} \) |
| 97 | \( 1 - 9454010975 T + p^{10} T^{2} \) |
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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
−18.67704884294538686201447588442, −15.85372630837309850941523118914, −15.39781714429878451741999206620, −13.95776866955299876862282975765, −12.31478968617794507728231101048, −10.25817769474062634910884974127, −8.278766239970351249717250305101, −7.38345213352865590490252353772, −3.66171223544846322147813083644, −2.21065338197171104042160731297,
2.21065338197171104042160731297, 3.66171223544846322147813083644, 7.38345213352865590490252353772, 8.278766239970351249717250305101, 10.25817769474062634910884974127, 12.31478968617794507728231101048, 13.95776866955299876862282975765, 15.39781714429878451741999206620, 15.85372630837309850941523118914, 18.67704884294538686201447588442
