| L(s) = 1 | + 2-s − 8-s − 13-s − 16-s − 23-s + 25-s − 26-s + 29-s − 31-s + 41-s − 46-s + 47-s + 49-s + 50-s + 58-s − 2·59-s − 62-s + 64-s + 71-s − 73-s + 82-s + 94-s + 98-s − 2·101-s + 104-s − 2·118-s + ⋯ |
| L(s) = 1 | + 2-s − 8-s − 13-s − 16-s − 23-s + 25-s − 26-s + 29-s − 31-s + 41-s − 46-s + 47-s + 49-s + 50-s + 58-s − 2·59-s − 62-s + 64-s + 71-s − 73-s + 82-s + 94-s + 98-s − 2·101-s + 104-s − 2·118-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9571325096\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9571325096\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - T + T^{2} \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 + T )^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−12.51999818433155963896736953468, −12.15059223284759182942833139437, −10.84905734670079491932064244344, −9.693253185299005466803205222451, −8.733275603796847118193368196479, −7.41279124382115205472213240668, −6.16971551051508484219371295762, −5.09954582736115586470949868349, −4.10638738448620025941358149709, −2.70040131053722579925112536657,
2.70040131053722579925112536657, 4.10638738448620025941358149709, 5.09954582736115586470949868349, 6.16971551051508484219371295762, 7.41279124382115205472213240668, 8.733275603796847118193368196479, 9.693253185299005466803205222451, 10.84905734670079491932064244344, 12.15059223284759182942833139437, 12.51999818433155963896736953468
