| L(s) = 1 | − 3·5-s + 4·7-s + 3·13-s − 3·17-s + 4·19-s + 8·23-s + 4·25-s + 5·29-s − 4·31-s − 12·35-s + 11·37-s − 7·41-s − 12·43-s + 8·47-s + 9·49-s + 53-s + 4·59-s − 2·61-s − 9·65-s + 4·67-s − 12·71-s + 10·73-s + 8·79-s − 4·83-s + 9·85-s − 3·89-s + 12·91-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 1.51·7-s + 0.832·13-s − 0.727·17-s + 0.917·19-s + 1.66·23-s + 4/5·25-s + 0.928·29-s − 0.718·31-s − 2.02·35-s + 1.80·37-s − 1.09·41-s − 1.82·43-s + 1.16·47-s + 9/7·49-s + 0.137·53-s + 0.520·59-s − 0.256·61-s − 1.11·65-s + 0.488·67-s − 1.42·71-s + 1.17·73-s + 0.900·79-s − 0.439·83-s + 0.976·85-s − 0.317·89-s + 1.25·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.107563639\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.107563639\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−7.82632753233195876356858866929, −7.20903365569409452502095833573, −6.60058503098257872425400535970, −5.48300649622903606001734853369, −4.87390661141273675062174989320, −4.31176367533593816489669194041, −3.57296668771543995429582891415, −2.75537476022977639105674075713, −1.55839762960904276906976011824, −0.76488839569876568582886783571,
0.76488839569876568582886783571, 1.55839762960904276906976011824, 2.75537476022977639105674075713, 3.57296668771543995429582891415, 4.31176367533593816489669194041, 4.87390661141273675062174989320, 5.48300649622903606001734853369, 6.60058503098257872425400535970, 7.20903365569409452502095833573, 7.82632753233195876356858866929
