| L(s) = 1 | + 2-s + 4-s − 4·7-s + 8-s − 11-s + 13-s − 4·14-s + 16-s − 17-s + 2·19-s − 22-s − 5·25-s + 26-s − 4·28-s + 6·29-s + 8·31-s + 32-s − 34-s + 2·37-s + 2·38-s + 6·41-s + 2·43-s − 44-s + 6·47-s + 9·49-s − 5·50-s + 52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.51·7-s + 0.353·8-s − 0.301·11-s + 0.277·13-s − 1.06·14-s + 1/4·16-s − 0.242·17-s + 0.458·19-s − 0.213·22-s − 25-s + 0.196·26-s − 0.755·28-s + 1.11·29-s + 1.43·31-s + 0.176·32-s − 0.171·34-s + 0.328·37-s + 0.324·38-s + 0.937·41-s + 0.304·43-s − 0.150·44-s + 0.875·47-s + 9/7·49-s − 0.707·50-s + 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43758 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43758 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.545190516\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.545190516\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−14.74497697960758, −13.83862813093614, −13.65314730953398, −13.28812537910883, −12.60968925645309, −12.13705122949879, −11.84902029346210, −11.04353600686319, −10.42512087612589, −10.14544132864427, −9.325056595995177, −9.145037267764706, −8.124811868098347, −7.731454723604603, −7.005380505940089, −6.455019179128464, −6.015085193385169, −5.617038189673252, −4.612010785350366, −4.285120909521553, −3.468586775193149, −2.907439276027864, −2.530813505163291, −1.439233206032988, −0.5101520480607770,
0.5101520480607770, 1.439233206032988, 2.530813505163291, 2.907439276027864, 3.468586775193149, 4.285120909521553, 4.612010785350366, 5.617038189673252, 6.015085193385169, 6.455019179128464, 7.005380505940089, 7.731454723604603, 8.124811868098347, 9.145037267764706, 9.325056595995177, 10.14544132864427, 10.42512087612589, 11.04353600686319, 11.84902029346210, 12.13705122949879, 12.60968925645309, 13.28812537910883, 13.65314730953398, 13.83862813093614, 14.74497697960758
