| L(s) = 1 | − 3-s − 7-s − 2·9-s + 3·11-s − 3·13-s + 3·17-s + 2·19-s + 21-s + 23-s + 5·27-s − 3·29-s − 4·31-s − 3·33-s − 8·37-s + 3·39-s − 10·41-s − 4·43-s + 13·47-s + 49-s − 3·51-s − 2·57-s + 8·59-s + 10·61-s + 2·63-s + 6·67-s − 69-s + 6·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s − 2/3·9-s + 0.904·11-s − 0.832·13-s + 0.727·17-s + 0.458·19-s + 0.218·21-s + 0.208·23-s + 0.962·27-s − 0.557·29-s − 0.718·31-s − 0.522·33-s − 1.31·37-s + 0.480·39-s − 1.56·41-s − 0.609·43-s + 1.89·47-s + 1/7·49-s − 0.420·51-s − 0.264·57-s + 1.04·59-s + 1.28·61-s + 0.251·63-s + 0.733·67-s − 0.120·69-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−12.98724953422547, −12.33849026785567, −12.06609472451704, −11.88276115786824, −11.20589067063152, −10.83492629764056, −10.30366663827862, −9.761160385754838, −9.464630465648435, −8.879300384289586, −8.441386250115853, −7.941004024691851, −7.219509341231887, −6.809124175391533, −6.632508612887319, −5.668840228953267, −5.472068859947538, −5.150359603113008, −4.364323461806832, −3.671346916004257, −3.430135280976496, −2.688856111442719, −2.099283890894685, −1.373511268207863, −0.6802408535314143, 0,
0.6802408535314143, 1.373511268207863, 2.099283890894685, 2.688856111442719, 3.430135280976496, 3.671346916004257, 4.364323461806832, 5.150359603113008, 5.472068859947538, 5.668840228953267, 6.632508612887319, 6.809124175391533, 7.219509341231887, 7.941004024691851, 8.441386250115853, 8.879300384289586, 9.464630465648435, 9.761160385754838, 10.30366663827862, 10.83492629764056, 11.20589067063152, 11.88276115786824, 12.06609472451704, 12.33849026785567, 12.98724953422547
