| L(s) = 1 | + 3-s − 2·9-s + 3·11-s − 13-s − 3·17-s − 2·19-s + 6·23-s − 5·27-s − 9·29-s − 8·31-s + 3·33-s + 10·37-s − 39-s − 2·43-s − 3·47-s − 3·51-s − 2·57-s − 12·59-s − 8·61-s − 8·67-s + 6·69-s + 14·73-s + 5·79-s + 81-s − 12·83-s − 9·87-s − 12·89-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2/3·9-s + 0.904·11-s − 0.277·13-s − 0.727·17-s − 0.458·19-s + 1.25·23-s − 0.962·27-s − 1.67·29-s − 1.43·31-s + 0.522·33-s + 1.64·37-s − 0.160·39-s − 0.304·43-s − 0.437·47-s − 0.420·51-s − 0.264·57-s − 1.56·59-s − 1.02·61-s − 0.977·67-s + 0.722·69-s + 1.63·73-s + 0.562·79-s + 1/9·81-s − 1.31·83-s − 0.964·87-s − 1.27·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−7.87248618599210160617864804548, −7.30396229458919554298380666065, −6.45992707641937946655967094066, −5.79449036400094857614726651847, −4.89997150704600145968375300343, −4.03340358569054504766377905041, −3.30687753040025310005397419375, −2.43854616662308710536414202448, −1.53894364511776905483657034012, 0,
1.53894364511776905483657034012, 2.43854616662308710536414202448, 3.30687753040025310005397419375, 4.03340358569054504766377905041, 4.89997150704600145968375300343, 5.79449036400094857614726651847, 6.45992707641937946655967094066, 7.30396229458919554298380666065, 7.87248618599210160617864804548
