| 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
| \(L(1)\) |
\(\approx\) |
\(1.574587015\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.574587015\) |
| \(L(\frac{3}{2})\) |
|
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
−8.882077076480821944930671478832, −7.86701473712700645017156466164, −7.32365997371505449239620211018, −6.22498866434346445412413636733, −5.35738096819619182003575665814, −5.22011328331955363777469451144, −3.77711727986427112393690247571, −3.42093209208388152186438852793, −2.40158712036426325906396194295, −0.67685474994955480555737946287,
0.67685474994955480555737946287, 2.40158712036426325906396194295, 3.42093209208388152186438852793, 3.77711727986427112393690247571, 5.22011328331955363777469451144, 5.35738096819619182003575665814, 6.22498866434346445412413636733, 7.32365997371505449239620211018, 7.86701473712700645017156466164, 8.882077076480821944930671478832
