| L(s) = 1 | + 7-s − 3·11-s + 13-s + 17-s + 8·19-s + 4·23-s + 7·29-s − 31-s + 4·37-s + 6·41-s + 12·43-s + 3·47-s − 6·49-s − 5·53-s − 9·59-s + 5·61-s + 11·67-s + 8·71-s − 3·77-s + 8·79-s + 7·83-s + 8·89-s + 91-s + 6·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 0.904·11-s + 0.277·13-s + 0.242·17-s + 1.83·19-s + 0.834·23-s + 1.29·29-s − 0.179·31-s + 0.657·37-s + 0.937·41-s + 1.82·43-s + 0.437·47-s − 6/7·49-s − 0.686·53-s − 1.17·59-s + 0.640·61-s + 1.34·67-s + 0.949·71-s − 0.341·77-s + 0.900·79-s + 0.768·83-s + 0.847·89-s + 0.104·91-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.147517737\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.147517737\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−14.46595804053177, −14.02154442220595, −13.76759563773794, −12.89104709292285, −12.70445888882462, −11.99311783131051, −11.47089371440272, −10.94067298599115, −10.57145890335878, −9.862094798857361, −9.348402655924650, −8.944196491387996, −7.982600038069065, −7.851483186516089, −7.298606450402560, −6.523426012998334, −5.972310506262881, −5.168438967506147, −5.034646259579428, −4.194641935483963, −3.420627182453691, −2.838333988672516, −2.276087781008016, −1.176120481189265, −0.7343524184532013,
0.7343524184532013, 1.176120481189265, 2.276087781008016, 2.838333988672516, 3.420627182453691, 4.194641935483963, 5.034646259579428, 5.168438967506147, 5.972310506262881, 6.523426012998334, 7.298606450402560, 7.851483186516089, 7.982600038069065, 8.944196491387996, 9.348402655924650, 9.862094798857361, 10.57145890335878, 10.94067298599115, 11.47089371440272, 11.99311783131051, 12.70445888882462, 12.89104709292285, 13.76759563773794, 14.02154442220595, 14.46595804053177
