| L(s) = 1 | + 7-s − 11-s + 4·17-s − 4·19-s − 6·23-s − 6·29-s − 8·37-s − 2·41-s − 8·43-s − 8·47-s + 49-s − 2·53-s − 4·59-s + 2·61-s + 10·67-s + 12·71-s + 10·73-s − 77-s − 12·79-s + 12·83-s − 8·89-s − 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 0.301·11-s + 0.970·17-s − 0.917·19-s − 1.25·23-s − 1.11·29-s − 1.31·37-s − 0.312·41-s − 1.21·43-s − 1.16·47-s + 1/7·49-s − 0.274·53-s − 0.520·59-s + 0.256·61-s + 1.22·67-s + 1.42·71-s + 1.17·73-s − 0.113·77-s − 1.35·79-s + 1.31·83-s − 0.847·89-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.046469056\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.046469056\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−13.50758018691395, −12.72330202603529, −12.61101772082330, −12.01668306973216, −11.41329040748925, −11.14648364858183, −10.47316681210064, −9.998367570039904, −9.749135690987388, −8.992552106523948, −8.467882927786271, −8.008986172207441, −7.732446860162354, −6.939932809353468, −6.560104162802490, −5.920499737537502, −5.360440774301935, −4.993290511238527, −4.320043837496429, −3.581951722350137, −3.405969947333689, −2.375359509876234, −1.918796100390614, −1.346951036970850, −0.2965011989865731,
0.2965011989865731, 1.346951036970850, 1.918796100390614, 2.375359509876234, 3.405969947333689, 3.581951722350137, 4.320043837496429, 4.993290511238527, 5.360440774301935, 5.920499737537502, 6.560104162802490, 6.939932809353468, 7.732446860162354, 8.008986172207441, 8.467882927786271, 8.992552106523948, 9.749135690987388, 9.998367570039904, 10.47316681210064, 11.14648364858183, 11.41329040748925, 12.01668306973216, 12.61101772082330, 12.72330202603529, 13.50758018691395
