| L(s) = 1 | − 113·3-s + 256·4-s + 1.15e3·5-s + 6.20e3·9-s + 1.46e4·11-s − 2.89e4·12-s − 1.30e5·15-s + 6.55e4·16-s + 2.94e5·20-s − 5.31e5·23-s + 9.34e5·25-s + 3.98e4·27-s − 1.54e6·31-s − 1.65e6·33-s + 1.58e6·36-s + 7.16e5·37-s + 3.74e6·44-s + 7.14e6·45-s − 6.08e6·47-s − 7.40e6·48-s + 5.76e6·49-s − 1.52e7·53-s + 1.68e7·55-s − 4.10e6·59-s − 3.32e7·60-s + 1.67e7·64-s + 1.98e7·67-s + ⋯ |
| L(s) = 1 | − 1.39·3-s + 4-s + 1.84·5-s + 0.946·9-s + 11-s − 1.39·12-s − 2.56·15-s + 16-s + 1.84·20-s − 1.90·23-s + 2.39·25-s + 0.0750·27-s − 1.66·31-s − 1.39·33-s + 0.946·36-s + 0.382·37-s + 44-s + 1.74·45-s − 1.24·47-s − 1.39·48-s + 49-s − 1.93·53-s + 1.84·55-s − 0.338·59-s − 2.56·60-s + 64-s + 0.982·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.573596594\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.573596594\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - p^{4} T \) |
| good | 2 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 3 | \( 1 + 113 T + p^{8} T^{2} \) |
| 5 | \( 1 - 1151 T + p^{8} T^{2} \) |
| 7 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 13 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 17 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 19 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 23 | \( 1 + 531793 T + p^{8} T^{2} \) |
| 29 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 31 | \( 1 + 1541233 T + p^{8} T^{2} \) |
| 37 | \( 1 - 716447 T + p^{8} T^{2} \) |
| 41 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 43 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 47 | \( 1 + 6080638 T + p^{8} T^{2} \) |
| 53 | \( 1 + 15265438 T + p^{8} T^{2} \) |
| 59 | \( 1 + 4101553 T + p^{8} T^{2} \) |
| 61 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 67 | \( 1 - 19806767 T + p^{8} T^{2} \) |
| 71 | \( 1 - 7043087 T + p^{8} T^{2} \) |
| 73 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 79 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 83 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 89 | \( 1 + 84100993 T + p^{8} T^{2} \) |
| 97 | \( 1 + 81155713 T + p^{8} T^{2} \) |
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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
−18.12727586097598137412742990628, −17.16262785141183334089759385559, −16.35260379416012785712797570839, −14.28880054289253718373030609722, −12.46276776345294754052969292615, −11.09484803513171732799778949987, −9.828176695462170004620426610180, −6.53233859618543442273249017838, −5.70223112857235470281460369734, −1.69504676247389347822626893512,
1.69504676247389347822626893512, 5.70223112857235470281460369734, 6.53233859618543442273249017838, 9.828176695462170004620426610180, 11.09484803513171732799778949987, 12.46276776345294754052969292615, 14.28880054289253718373030609722, 16.35260379416012785712797570839, 17.16262785141183334089759385559, 18.12727586097598137412742990628
