| L(s) = 1 | + 1.41·2-s − 1.58·5-s + 7-s − 2.82·8-s − 2.24·10-s − 4.24·11-s − 13-s + 1.41·14-s − 4.00·16-s − 1.41·17-s − 7.24·19-s − 6·22-s + 5.82·23-s − 2.48·25-s − 1.41·26-s − 0.171·29-s + 3.24·31-s − 2.00·34-s − 1.58·35-s + 2.24·37-s − 10.2·38-s + 4.48·40-s − 8.82·41-s − 5·43-s + ⋯ |
| L(s) = 1 | + 1.00·2-s − 0.709·5-s + 0.377·7-s − 0.999·8-s − 0.709·10-s − 1.27·11-s − 0.277·13-s + 0.377·14-s − 1.00·16-s − 0.342·17-s − 1.66·19-s − 1.27·22-s + 1.21·23-s − 0.497·25-s − 0.277·26-s − 0.0318·29-s + 0.582·31-s − 0.342·34-s − 0.268·35-s + 0.368·37-s − 1.66·38-s + 0.709·40-s − 1.37·41-s − 0.762·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 1.41T + 2T^{2} \) |
| 5 | \( 1 + 1.58T + 5T^{2} \) |
| 11 | \( 1 + 4.24T + 11T^{2} \) |
| 17 | \( 1 + 1.41T + 17T^{2} \) |
| 19 | \( 1 + 7.24T + 19T^{2} \) |
| 23 | \( 1 - 5.82T + 23T^{2} \) |
| 29 | \( 1 + 0.171T + 29T^{2} \) |
| 31 | \( 1 - 3.24T + 31T^{2} \) |
| 37 | \( 1 - 2.24T + 37T^{2} \) |
| 41 | \( 1 + 8.82T + 41T^{2} \) |
| 43 | \( 1 + 5T + 43T^{2} \) |
| 47 | \( 1 + 1.58T + 47T^{2} \) |
| 53 | \( 1 - 0.171T + 53T^{2} \) |
| 59 | \( 1 + 0.343T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 14.4T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 + 9.24T + 73T^{2} \) |
| 79 | \( 1 - 15.4T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 + 1.58T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
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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
−9.948518543062624110733216325085, −8.716578056626868847548415981819, −8.190874997175287745470873021242, −7.12490428582570049174956945144, −6.11525822107720018812669680046, −5.02539958341989302646865844568, −4.52882069293821994846404005083, −3.45550186273424292901882204586, −2.38103717333951279595775291844, 0,
2.38103717333951279595775291844, 3.45550186273424292901882204586, 4.52882069293821994846404005083, 5.02539958341989302646865844568, 6.11525822107720018812669680046, 7.12490428582570049174956945144, 8.190874997175287745470873021242, 8.716578056626868847548415981819, 9.948518543062624110733216325085
