| L(s) = 1 | − 2·2-s + 4·4-s + 5·5-s + 8·7-s − 8·8-s − 10·10-s + 40·11-s − 13·13-s − 16·14-s + 16·16-s − 10·17-s + 20·20-s − 80·22-s + 180·23-s + 25·25-s + 26·26-s + 32·28-s − 22·29-s − 144·31-s − 32·32-s + 20·34-s + 40·35-s + 34·37-s − 40·40-s + 502·41-s − 76·43-s + 160·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.431·7-s − 0.353·8-s − 0.316·10-s + 1.09·11-s − 0.277·13-s − 0.305·14-s + 1/4·16-s − 0.142·17-s + 0.223·20-s − 0.775·22-s + 1.63·23-s + 1/5·25-s + 0.196·26-s + 0.215·28-s − 0.140·29-s − 0.834·31-s − 0.176·32-s + 0.100·34-s + 0.193·35-s + 0.151·37-s − 0.158·40-s + 1.91·41-s − 0.269·43-s + 0.548·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.945221800\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.945221800\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 13 | \( 1 + p T \) |
| good | 7 | \( 1 - 8 T + p^{3} T^{2} \) |
| 11 | \( 1 - 40 T + p^{3} T^{2} \) |
| 17 | \( 1 + 10 T + p^{3} T^{2} \) |
| 19 | \( 1 + p^{3} T^{2} \) |
| 23 | \( 1 - 180 T + p^{3} T^{2} \) |
| 29 | \( 1 + 22 T + p^{3} T^{2} \) |
| 31 | \( 1 + 144 T + p^{3} T^{2} \) |
| 37 | \( 1 - 34 T + p^{3} T^{2} \) |
| 41 | \( 1 - 502 T + p^{3} T^{2} \) |
| 43 | \( 1 + 76 T + p^{3} T^{2} \) |
| 47 | \( 1 - 168 T + p^{3} T^{2} \) |
| 53 | \( 1 - 422 T + p^{3} T^{2} \) |
| 59 | \( 1 + 104 T + p^{3} T^{2} \) |
| 61 | \( 1 + 82 T + p^{3} T^{2} \) |
| 67 | \( 1 + 540 T + p^{3} T^{2} \) |
| 71 | \( 1 + 512 T + p^{3} T^{2} \) |
| 73 | \( 1 - 622 T + p^{3} T^{2} \) |
| 79 | \( 1 - 104 T + p^{3} T^{2} \) |
| 83 | \( 1 + 348 T + p^{3} T^{2} \) |
| 89 | \( 1 - 286 T + p^{3} T^{2} \) |
| 97 | \( 1 - 494 T + p^{3} T^{2} \) |
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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
−9.117200358709171689216198072317, −8.971345262199564415383077641613, −7.73885370838450580920039435136, −7.03573666455482084024534028592, −6.20608400392910791468291600984, −5.24810324439319971807497319114, −4.17051020188263735636953728832, −2.92894795138904157930196319883, −1.79567424969206616362807599367, −0.839423849752883350041762224895,
0.839423849752883350041762224895, 1.79567424969206616362807599367, 2.92894795138904157930196319883, 4.17051020188263735636953728832, 5.24810324439319971807497319114, 6.20608400392910791468291600984, 7.03573666455482084024534028592, 7.73885370838450580920039435136, 8.971345262199564415383077641613, 9.117200358709171689216198072317
