| L(s) = 1 | − 3·3-s − 5·7-s + 6·9-s − 4·13-s − 17-s + 5·19-s + 15·21-s − 4·23-s − 9·27-s − 5·29-s − 31-s + 3·37-s + 12·39-s + 2·41-s + 8·43-s − 6·47-s + 18·49-s + 3·51-s + 11·53-s − 15·57-s + 5·61-s − 30·63-s − 4·67-s + 12·69-s − 15·71-s − 6·73-s + 8·79-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.88·7-s + 2·9-s − 1.10·13-s − 0.242·17-s + 1.14·19-s + 3.27·21-s − 0.834·23-s − 1.73·27-s − 0.928·29-s − 0.179·31-s + 0.493·37-s + 1.92·39-s + 0.312·41-s + 1.21·43-s − 0.875·47-s + 18/7·49-s + 0.420·51-s + 1.51·53-s − 1.98·57-s + 0.640·61-s − 3.77·63-s − 0.488·67-s + 1.44·69-s − 1.78·71-s − 0.702·73-s + 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.20736544096324, −12.68792640326714, −12.32312318143835, −12.00847698790227, −11.51136059807122, −11.08155289166209, −10.34848736119563, −10.15033044181888, −9.696761007060508, −9.315240262273409, −8.817244203179619, −7.661471410676423, −7.508972534220081, −6.875206794868106, −6.581376525840675, −5.931799032175644, −5.661567667916910, −5.252141233891732, −4.477752317600936, −4.054580327580775, −3.439124886686015, −2.764660303984059, −2.174304865038498, −1.207504021050486, −0.5077234138382241, 0,
0.5077234138382241, 1.207504021050486, 2.174304865038498, 2.764660303984059, 3.439124886686015, 4.054580327580775, 4.477752317600936, 5.252141233891732, 5.661567667916910, 5.931799032175644, 6.581376525840675, 6.875206794868106, 7.508972534220081, 7.661471410676423, 8.817244203179619, 9.315240262273409, 9.696761007060508, 10.15033044181888, 10.34848736119563, 11.08155289166209, 11.51136059807122, 12.00847698790227, 12.32312318143835, 12.68792640326714, 13.20736544096324
