| L(s) = 1 | − 0.486·3-s − 3.63·7-s − 2.76·9-s − 2.79·11-s − 2.86·13-s − 1.17·17-s + 19-s + 1.76·21-s + 0.617·23-s + 2.80·27-s − 4.96·29-s + 0.745·31-s + 1.36·33-s − 8.23·37-s + 1.39·39-s + 9.98·41-s − 10.4·43-s − 5.07·47-s + 6.19·49-s + 0.571·51-s + 7.45·53-s − 0.486·57-s + 3.83·59-s + 11.2·61-s + 10.0·63-s − 6.10·67-s − 0.300·69-s + ⋯ |
| L(s) = 1 | − 0.280·3-s − 1.37·7-s − 0.921·9-s − 0.842·11-s − 0.794·13-s − 0.284·17-s + 0.229·19-s + 0.385·21-s + 0.128·23-s + 0.539·27-s − 0.922·29-s + 0.133·31-s + 0.236·33-s − 1.35·37-s + 0.223·39-s + 1.55·41-s − 1.60·43-s − 0.740·47-s + 0.885·49-s + 0.0800·51-s + 1.02·53-s − 0.0644·57-s + 0.499·59-s + 1.43·61-s + 1.26·63-s − 0.746·67-s − 0.0361·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5419413374\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5419413374\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 0.486T + 3T^{2} \) |
| 7 | \( 1 + 3.63T + 7T^{2} \) |
| 11 | \( 1 + 2.79T + 11T^{2} \) |
| 13 | \( 1 + 2.86T + 13T^{2} \) |
| 17 | \( 1 + 1.17T + 17T^{2} \) |
| 23 | \( 1 - 0.617T + 23T^{2} \) |
| 29 | \( 1 + 4.96T + 29T^{2} \) |
| 31 | \( 1 - 0.745T + 31T^{2} \) |
| 37 | \( 1 + 8.23T + 37T^{2} \) |
| 41 | \( 1 - 9.98T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 + 5.07T + 47T^{2} \) |
| 53 | \( 1 - 7.45T + 53T^{2} \) |
| 59 | \( 1 - 3.83T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 + 6.10T + 67T^{2} \) |
| 71 | \( 1 + 9.40T + 71T^{2} \) |
| 73 | \( 1 - 9.52T + 73T^{2} \) |
| 79 | \( 1 + 3.70T + 79T^{2} \) |
| 83 | \( 1 - 4.66T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 + 0.629T + 97T^{2} \) |
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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
−8.593222436890973096530346120742, −7.67694376092315392620037092141, −6.97151306292891256872763844374, −6.26759576648153510219174166721, −5.51143912788659048075711805819, −4.93133487196376022304935597818, −3.69035037313820857998537092097, −2.98611152398659657223707538927, −2.20796277163223607968511295493, −0.39984940269767498431159631899,
0.39984940269767498431159631899, 2.20796277163223607968511295493, 2.98611152398659657223707538927, 3.69035037313820857998537092097, 4.93133487196376022304935597818, 5.51143912788659048075711805819, 6.26759576648153510219174166721, 6.97151306292891256872763844374, 7.67694376092315392620037092141, 8.593222436890973096530346120742
