| L(s) = 1 | − 3-s − 4·7-s + 9-s − 11-s + 2·13-s + 17-s + 6·19-s + 4·21-s + 8·23-s − 27-s + 2·29-s − 2·31-s + 33-s − 2·37-s − 2·39-s − 2·41-s + 4·43-s − 6·47-s + 9·49-s − 51-s − 6·53-s − 6·57-s − 4·59-s − 2·61-s − 4·63-s + 12·67-s − 8·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 0.242·17-s + 1.37·19-s + 0.872·21-s + 1.66·23-s − 0.192·27-s + 0.371·29-s − 0.359·31-s + 0.174·33-s − 0.328·37-s − 0.320·39-s − 0.312·41-s + 0.609·43-s − 0.875·47-s + 9/7·49-s − 0.140·51-s − 0.824·53-s − 0.794·57-s − 0.520·59-s − 0.256·61-s − 0.503·63-s + 1.46·67-s − 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.753623735\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.753623735\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−12.98259488138120, −12.51026433865537, −12.13118094863825, −11.51608616910490, −11.07257665281096, −10.68376491203427, −10.13375079724086, −9.674293868878586, −9.264317214281884, −8.974381678891675, −8.117491276378126, −7.721567246974771, −7.089864729076366, −6.629349236603840, −6.400315357216862, −5.712311672603866, −5.167761614075374, −4.960581178103165, −4.011398605881690, −3.491470817928823, −3.116695601511594, −2.618125211563567, −1.672015216472826, −0.9514902876746203, −0.4703306372279048,
0.4703306372279048, 0.9514902876746203, 1.672015216472826, 2.618125211563567, 3.116695601511594, 3.491470817928823, 4.011398605881690, 4.960581178103165, 5.167761614075374, 5.712311672603866, 6.400315357216862, 6.629349236603840, 7.089864729076366, 7.721567246974771, 8.117491276378126, 8.974381678891675, 9.264317214281884, 9.674293868878586, 10.13375079724086, 10.68376491203427, 11.07257665281096, 11.51608616910490, 12.13118094863825, 12.51026433865537, 12.98259488138120
