| L(s) = 1 | − 2·3-s + 6·7-s − 23·9-s − 60·11-s − 50·13-s + 30·17-s − 40·19-s − 12·21-s + 178·23-s + 100·27-s + 166·29-s − 20·31-s + 120·33-s − 10·37-s + 100·39-s − 250·41-s + 142·43-s + 214·47-s − 307·49-s − 60·51-s − 490·53-s + 80·57-s + 800·59-s + 250·61-s − 138·63-s − 774·67-s − 356·69-s + ⋯ |
| L(s) = 1 | − 0.384·3-s + 0.323·7-s − 0.851·9-s − 1.64·11-s − 1.06·13-s + 0.428·17-s − 0.482·19-s − 0.124·21-s + 1.61·23-s + 0.712·27-s + 1.06·29-s − 0.115·31-s + 0.633·33-s − 0.0444·37-s + 0.410·39-s − 0.952·41-s + 0.503·43-s + 0.664·47-s − 0.895·49-s − 0.164·51-s − 1.26·53-s + 0.185·57-s + 1.76·59-s + 0.524·61-s − 0.275·63-s − 1.41·67-s − 0.621·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.091691691\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.091691691\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p^{3} T^{2} \) |
| 7 | \( 1 - 6 T + p^{3} T^{2} \) |
| 11 | \( 1 + 60 T + p^{3} T^{2} \) |
| 13 | \( 1 + 50 T + p^{3} T^{2} \) |
| 17 | \( 1 - 30 T + p^{3} T^{2} \) |
| 19 | \( 1 + 40 T + p^{3} T^{2} \) |
| 23 | \( 1 - 178 T + p^{3} T^{2} \) |
| 29 | \( 1 - 166 T + p^{3} T^{2} \) |
| 31 | \( 1 + 20 T + p^{3} T^{2} \) |
| 37 | \( 1 + 10 T + p^{3} T^{2} \) |
| 41 | \( 1 + 250 T + p^{3} T^{2} \) |
| 43 | \( 1 - 142 T + p^{3} T^{2} \) |
| 47 | \( 1 - 214 T + p^{3} T^{2} \) |
| 53 | \( 1 + 490 T + p^{3} T^{2} \) |
| 59 | \( 1 - 800 T + p^{3} T^{2} \) |
| 61 | \( 1 - 250 T + p^{3} T^{2} \) |
| 67 | \( 1 + 774 T + p^{3} T^{2} \) |
| 71 | \( 1 + 100 T + p^{3} T^{2} \) |
| 73 | \( 1 - 230 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1320 T + p^{3} T^{2} \) |
| 83 | \( 1 - 982 T + p^{3} T^{2} \) |
| 89 | \( 1 - 874 T + p^{3} T^{2} \) |
| 97 | \( 1 - 310 T + p^{3} T^{2} \) |
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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
−10.08421165210963073679623468747, −8.950415919407380678132044503976, −8.123621457200789690700512275096, −7.38136997373497635068579341322, −6.32378023986708973375165579652, −5.14530881383801815440747322818, −4.92749321527704198485452966509, −3.15142490379693847056066890327, −2.33789161253966981763205654397, −0.56866515580141363377599688471,
0.56866515580141363377599688471, 2.33789161253966981763205654397, 3.15142490379693847056066890327, 4.92749321527704198485452966509, 5.14530881383801815440747322818, 6.32378023986708973375165579652, 7.38136997373497635068579341322, 8.123621457200789690700512275096, 8.950415919407380678132044503976, 10.08421165210963073679623468747
