| L(s) = 1 | − 2-s + 4-s + 1.23·5-s + 2.38·7-s − 8-s − 1.23·10-s + 3.47·13-s − 2.38·14-s + 16-s − 5·17-s + 5·19-s + 1.23·20-s − 2.85·23-s − 3.47·25-s − 3.47·26-s + 2.38·28-s − 5.85·29-s − 2.23·31-s − 32-s + 5·34-s + 2.94·35-s − 8.23·37-s − 5·38-s − 1.23·40-s − 3.85·41-s − 10.2·43-s + 2.85·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.552·5-s + 0.900·7-s − 0.353·8-s − 0.390·10-s + 0.962·13-s − 0.636·14-s + 0.250·16-s − 1.21·17-s + 1.14·19-s + 0.276·20-s − 0.595·23-s − 0.694·25-s − 0.680·26-s + 0.450·28-s − 1.08·29-s − 0.401·31-s − 0.176·32-s + 0.857·34-s + 0.497·35-s − 1.35·37-s − 0.811·38-s − 0.195·40-s − 0.601·41-s − 1.56·43-s + 0.420·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 1.23T + 5T^{2} \) |
| 7 | \( 1 - 2.38T + 7T^{2} \) |
| 13 | \( 1 - 3.47T + 13T^{2} \) |
| 17 | \( 1 + 5T + 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 + 2.85T + 23T^{2} \) |
| 29 | \( 1 + 5.85T + 29T^{2} \) |
| 31 | \( 1 + 2.23T + 31T^{2} \) |
| 37 | \( 1 + 8.23T + 37T^{2} \) |
| 41 | \( 1 + 3.85T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 + 4.70T + 47T^{2} \) |
| 53 | \( 1 - 5.32T + 53T^{2} \) |
| 59 | \( 1 + 9.61T + 59T^{2} \) |
| 61 | \( 1 - 6.70T + 61T^{2} \) |
| 67 | \( 1 + 13.3T + 67T^{2} \) |
| 71 | \( 1 + 9T + 71T^{2} \) |
| 73 | \( 1 + 0.708T + 73T^{2} \) |
| 79 | \( 1 + 4.70T + 79T^{2} \) |
| 83 | \( 1 + 3.94T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 + 8.47T + 97T^{2} \) |
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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
−7.68641965671700257370696154506, −7.11079527196877987433588230725, −6.27690967081554988224777997397, −5.63234449708281233978652282792, −4.92243494628335831897900102943, −3.93087942845616456330707941835, −3.08315913737671935000303591007, −1.79539961366855540900585859148, −1.60910358859866097935282352938, 0,
1.60910358859866097935282352938, 1.79539961366855540900585859148, 3.08315913737671935000303591007, 3.93087942845616456330707941835, 4.92243494628335831897900102943, 5.63234449708281233978652282792, 6.27690967081554988224777997397, 7.11079527196877987433588230725, 7.68641965671700257370696154506
