| L(s) = 1 | + 3.46·5-s − 1.73·7-s − 6·11-s + 5.19·13-s − 6·17-s + 5·19-s + 3.46·23-s + 6.99·25-s + 6.92·29-s + 3.46·31-s − 5.99·35-s − 1.73·37-s − 4·43-s − 3.46·47-s − 4·49-s + 6.92·53-s − 20.7·55-s + 6·59-s + 12.1·61-s + 18·65-s − 5·67-s − 13.8·71-s + 7·73-s + 10.3·77-s + 5.19·79-s − 20.7·85-s − 6·89-s + ⋯ |
| L(s) = 1 | + 1.54·5-s − 0.654·7-s − 1.80·11-s + 1.44·13-s − 1.45·17-s + 1.14·19-s + 0.722·23-s + 1.39·25-s + 1.28·29-s + 0.622·31-s − 1.01·35-s − 0.284·37-s − 0.609·43-s − 0.505·47-s − 0.571·49-s + 0.951·53-s − 2.80·55-s + 0.781·59-s + 1.55·61-s + 2.23·65-s − 0.610·67-s − 1.64·71-s + 0.819·73-s + 1.18·77-s + 0.584·79-s − 2.25·85-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.435392770\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.435392770\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 3.46T + 5T^{2} \) |
| 7 | \( 1 + 1.73T + 7T^{2} \) |
| 11 | \( 1 + 6T + 11T^{2} \) |
| 13 | \( 1 - 5.19T + 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 - 6.92T + 29T^{2} \) |
| 31 | \( 1 - 3.46T + 31T^{2} \) |
| 37 | \( 1 + 1.73T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 3.46T + 47T^{2} \) |
| 53 | \( 1 - 6.92T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 + 5T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 - 7T + 73T^{2} \) |
| 79 | \( 1 - 5.19T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
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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
−8.135420658767901557693414611396, −7.00022295981247855103029155121, −6.53755510805309005615863344993, −5.81954375228565035713703468579, −5.28390297596266538251686139610, −4.57619714800079311367462289861, −3.27840650698928353622550678462, −2.72477272404209041035507212980, −1.92330816828387369178197026313, −0.793777043354786015078295583481,
0.793777043354786015078295583481, 1.92330816828387369178197026313, 2.72477272404209041035507212980, 3.27840650698928353622550678462, 4.57619714800079311367462289861, 5.28390297596266538251686139610, 5.81954375228565035713703468579, 6.53755510805309005615863344993, 7.00022295981247855103029155121, 8.135420658767901557693414611396
