| L(s) = 1 | − 3·3-s + 6·9-s − 11-s − 6·13-s + 5·17-s − 2·19-s + 2·23-s − 9·27-s − 6·29-s − 3·31-s + 3·33-s − 5·37-s + 18·39-s + 6·41-s + 11·43-s − 47-s − 15·51-s + 53-s + 6·57-s − 15·59-s − 10·61-s + 10·67-s − 6·69-s + 10·71-s − 5·73-s − 17·79-s + 9·81-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 2·9-s − 0.301·11-s − 1.66·13-s + 1.21·17-s − 0.458·19-s + 0.417·23-s − 1.73·27-s − 1.11·29-s − 0.538·31-s + 0.522·33-s − 0.821·37-s + 2.88·39-s + 0.937·41-s + 1.67·43-s − 0.145·47-s − 2.10·51-s + 0.137·53-s + 0.794·57-s − 1.95·59-s − 1.28·61-s + 1.22·67-s − 0.722·69-s + 1.18·71-s − 0.585·73-s − 1.91·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4344322398\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4344322398\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + 17 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−14.36081483044774, −14.10407209533256, −13.09911443544585, −12.60574154677332, −12.44026188510153, −11.98131648135396, −11.22999557965067, −11.02014642832655, −10.35700575650020, −9.977317141855576, −9.446320295839420, −8.878855818636661, −7.841335162368083, −7.370328045903964, −7.195657641375988, −6.311906217695980, −5.776406761119590, −5.436044971156232, −4.810980900750129, −4.417994965463801, −3.592267323667051, −2.754535218586414, −1.951803757275118, −1.151212527472498, −0.2818882497981800,
0.2818882497981800, 1.151212527472498, 1.951803757275118, 2.754535218586414, 3.592267323667051, 4.417994965463801, 4.810980900750129, 5.436044971156232, 5.776406761119590, 6.311906217695980, 7.195657641375988, 7.370328045903964, 7.841335162368083, 8.878855818636661, 9.446320295839420, 9.977317141855576, 10.35700575650020, 11.02014642832655, 11.22999557965067, 11.98131648135396, 12.44026188510153, 12.60574154677332, 13.09911443544585, 14.10407209533256, 14.36081483044774
