| L(s) = 1 | − 2.08e4·3-s + 2.62e5·4-s − 3.06e6·5-s + 4.81e7·9-s − 2.35e9·11-s − 5.47e9·12-s + 6.39e10·15-s + 6.87e10·16-s − 8.03e11·20-s + 3.59e12·23-s + 5.57e12·25-s + 7.08e12·27-s + 2.66e13·31-s + 4.92e13·33-s + 1.26e13·36-s − 1.04e13·37-s − 6.18e14·44-s − 1.47e14·45-s − 2.11e15·47-s − 1.43e15·48-s + 1.62e15·49-s − 1.36e15·53-s + 7.22e15·55-s − 1.23e16·59-s + 1.67e16·60-s + 1.80e16·64-s + 3.76e16·67-s + ⋯ |
| L(s) = 1 | − 1.06·3-s + 4-s − 1.56·5-s + 0.124·9-s − 11-s − 1.06·12-s + 1.66·15-s + 16-s − 1.56·20-s + 1.99·23-s + 1.46·25-s + 0.928·27-s + 1.00·31-s + 1.06·33-s + 0.124·36-s − 0.0800·37-s − 44-s − 0.194·45-s − 1.88·47-s − 1.06·48-s + 49-s − 0.414·53-s + 1.56·55-s − 1.42·59-s + 1.66·60-s + 64-s + 1.38·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{19}{2})\) |
\(\approx\) |
\(0.9513278783\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9513278783\) |
| \(L(10)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + p^{9} T \) |
| good | 2 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 3 | \( 1 + 20870 T + p^{18} T^{2} \) |
| 5 | \( 1 + 3063526 T + p^{18} T^{2} \) |
| 7 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 13 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 17 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 19 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 23 | \( 1 - 3592942977890 T + p^{18} T^{2} \) |
| 29 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 31 | \( 1 - 26636854831058 T + p^{18} T^{2} \) |
| 37 | \( 1 + 10400449085350 T + p^{18} T^{2} \) |
| 41 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 43 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 47 | \( 1 + 2113576298457550 T + p^{18} T^{2} \) |
| 53 | \( 1 + 1367378362647430 T + p^{18} T^{2} \) |
| 59 | \( 1 + 12373964663300278 T + p^{18} T^{2} \) |
| 61 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 67 | \( 1 - 37614783844431290 T + p^{18} T^{2} \) |
| 71 | \( 1 - 91404547650956162 T + p^{18} T^{2} \) |
| 73 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 79 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 83 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 89 | \( 1 - 623295746335388018 T + p^{18} T^{2} \) |
| 97 | \( 1 + 1511803192851761470 T + p^{18} T^{2} \) |
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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
−15.98282965061972572331619812202, −15.15118531530360778909142974108, −12.53915799226184789673962052152, −11.47258335076925449606076909301, −10.76916654089980572576202783207, −8.006418059240926488514889632138, −6.74438991827731721149928356682, −5.03229244219874526414791117905, −3.05176215886430938277189603529, −0.67094384004634262423541116772,
0.67094384004634262423541116772, 3.05176215886430938277189603529, 5.03229244219874526414791117905, 6.74438991827731721149928356682, 8.006418059240926488514889632138, 10.76916654089980572576202783207, 11.47258335076925449606076909301, 12.53915799226184789673962052152, 15.15118531530360778909142974108, 15.98282965061972572331619812202
