L(s) = 1 | − 2-s − 4-s − 2·7-s + 3·8-s + 4·11-s + 13-s + 2·14-s − 16-s − 4·17-s + 6·19-s − 4·22-s − 26-s + 2·28-s + 4·29-s − 10·31-s − 5·32-s + 4·34-s + 2·37-s − 6·38-s + 6·41-s + 8·43-s − 4·44-s − 8·47-s − 3·49-s − 52-s − 4·53-s − 6·56-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.755·7-s + 1.06·8-s + 1.20·11-s + 0.277·13-s + 0.534·14-s − 1/4·16-s − 0.970·17-s + 1.37·19-s − 0.852·22-s − 0.196·26-s + 0.377·28-s + 0.742·29-s − 1.79·31-s − 0.883·32-s + 0.685·34-s + 0.328·37-s − 0.973·38-s + 0.937·41-s + 1.21·43-s − 0.603·44-s − 1.16·47-s − 3/7·49-s − 0.138·52-s − 0.549·53-s − 0.801·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9553241076\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9553241076\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−9.007441006651754297886939778829, −8.127287261470530597099570399956, −7.32235374032518873623110711846, −6.63345147905253550680928755279, −5.80924281505585277042661600069, −4.78697835354588764823687539378, −3.97557816773106922792443930782, −3.20198513482083274655459567901, −1.76669809243192085864580913398, −0.69407548081692426413856785850,
0.69407548081692426413856785850, 1.76669809243192085864580913398, 3.20198513482083274655459567901, 3.97557816773106922792443930782, 4.78697835354588764823687539378, 5.80924281505585277042661600069, 6.63345147905253550680928755279, 7.32235374032518873623110711846, 8.127287261470530597099570399956, 9.007441006651754297886939778829