L(s) = 1 | − 2-s − 4-s − 7-s + 3·8-s + 3·11-s + 7·13-s + 14-s − 16-s + 4·17-s + 3·19-s − 3·22-s − 7·26-s + 28-s − 2·29-s + 8·31-s − 5·32-s − 4·34-s − 4·37-s − 3·38-s + 11·41-s + 5·43-s − 3·44-s + 9·47-s + 49-s − 7·52-s − 3·53-s − 3·56-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.377·7-s + 1.06·8-s + 0.904·11-s + 1.94·13-s + 0.267·14-s − 1/4·16-s + 0.970·17-s + 0.688·19-s − 0.639·22-s − 1.37·26-s + 0.188·28-s − 0.371·29-s + 1.43·31-s − 0.883·32-s − 0.685·34-s − 0.657·37-s − 0.486·38-s + 1.71·41-s + 0.762·43-s − 0.452·44-s + 1.31·47-s + 1/7·49-s − 0.970·52-s − 0.412·53-s − 0.400·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.465521551\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.465521551\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 15 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.439127641977703771827087814336, −7.72785279165831200876264505139, −7.01087805293481845320253238111, −6.04225062588914423330171654760, −5.60362307113687757598906157956, −4.33575667623772700668880654561, −3.86004077446490800252010541769, −2.96852923192376208730019290665, −1.40802486646014090172011281371, −0.884928458861746948518336266283,
0.884928458861746948518336266283, 1.40802486646014090172011281371, 2.96852923192376208730019290665, 3.86004077446490800252010541769, 4.33575667623772700668880654561, 5.60362307113687757598906157956, 6.04225062588914423330171654760, 7.01087805293481845320253238111, 7.72785279165831200876264505139, 8.439127641977703771827087814336