| L(s) = 1 | − 2·2-s + 2·4-s − 5-s + 2·10-s + 2·11-s − 4·16-s − 6·17-s − 3·19-s − 2·20-s − 4·22-s − 3·23-s − 4·25-s − 3·29-s − 3·31-s + 8·32-s + 12·34-s + 6·37-s + 6·38-s − 10·41-s − 43-s + 4·44-s + 6·46-s − 11·47-s + 8·50-s + 9·53-s − 2·55-s + 6·58-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 0.447·5-s + 0.632·10-s + 0.603·11-s − 16-s − 1.45·17-s − 0.688·19-s − 0.447·20-s − 0.852·22-s − 0.625·23-s − 4/5·25-s − 0.557·29-s − 0.538·31-s + 1.41·32-s + 2.05·34-s + 0.986·37-s + 0.973·38-s − 1.56·41-s − 0.152·43-s + 0.603·44-s + 0.884·46-s − 1.60·47-s + 1.13·50-s + 1.23·53-s − 0.269·55-s + 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−14.81176762710047, −14.02907528282948, −13.53846404971233, −13.08470465077026, −12.54487323570332, −11.70164492196164, −11.41084112944444, −11.17291438786659, −10.26283939785205, −10.12559549961862, −9.433404790621302, −8.944094295794681, −8.515702860672844, −8.162000576806814, −7.433931795699814, −7.103543891397918, −6.438866180243405, −6.059394384291749, −5.124478856791953, −4.396034300076078, −4.067586984587048, −3.318142764542714, −2.319212601597863, −1.902165209807776, −1.218813014319785, 0, 0,
1.218813014319785, 1.902165209807776, 2.319212601597863, 3.318142764542714, 4.067586984587048, 4.396034300076078, 5.124478856791953, 6.059394384291749, 6.438866180243405, 7.103543891397918, 7.433931795699814, 8.162000576806814, 8.515702860672844, 8.944094295794681, 9.433404790621302, 10.12559549961862, 10.26283939785205, 11.17291438786659, 11.41084112944444, 11.70164492196164, 12.54487323570332, 13.08470465077026, 13.53846404971233, 14.02907528282948, 14.81176762710047
