| L(s) = 1 | + 4-s − 5-s + 7-s + 11-s − 13-s + 16-s − 17-s − 20-s + 2·23-s + 25-s + 28-s − 35-s − 2·37-s + 41-s + 44-s − 52-s − 53-s − 55-s + 59-s − 61-s + 64-s + 65-s + 67-s − 68-s + 71-s + 73-s + 77-s + ⋯ |
| L(s) = 1 | + 4-s − 5-s + 7-s + 11-s − 13-s + 16-s − 17-s − 20-s + 2·23-s + 25-s + 28-s − 35-s − 2·37-s + 41-s + 44-s − 52-s − 53-s − 55-s + 59-s − 61-s + 64-s + 65-s + 67-s − 68-s + 71-s + 73-s + 77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.376635040\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.376635040\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 + T )^{2} \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 - T + T^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 - T + T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 + T )^{2} \) |
| 97 | \( 1 - T + T^{2} \) |
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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
−9.357559567388110791394827368380, −8.604796076216358709738788757036, −7.82344084203355150156896751062, −7.00614434458170755397437305887, −6.71858072519163120399742659552, −5.27811517067648765123857363612, −4.58882145646556418430482264085, −3.56050049588852436746821711658, −2.52166829627096639404976530099, −1.35966544841404163128080562497,
1.35966544841404163128080562497, 2.52166829627096639404976530099, 3.56050049588852436746821711658, 4.58882145646556418430482264085, 5.27811517067648765123857363612, 6.71858072519163120399742659552, 7.00614434458170755397437305887, 7.82344084203355150156896751062, 8.604796076216358709738788757036, 9.357559567388110791394827368380
