| L(s) = 1 | − 3-s + 4·5-s + 7-s − 2·9-s + 3·11-s + 3·13-s − 4·15-s − 4·17-s − 21-s − 3·23-s + 11·25-s + 5·27-s + 10·29-s + 10·31-s − 3·33-s + 4·35-s − 2·37-s − 3·39-s + 2·41-s − 5·43-s − 8·45-s + 11·47-s − 6·49-s + 4·51-s + 14·53-s + 12·55-s + 4·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.78·5-s + 0.377·7-s − 2/3·9-s + 0.904·11-s + 0.832·13-s − 1.03·15-s − 0.970·17-s − 0.218·21-s − 0.625·23-s + 11/5·25-s + 0.962·27-s + 1.85·29-s + 1.79·31-s − 0.522·33-s + 0.676·35-s − 0.328·37-s − 0.480·39-s + 0.312·41-s − 0.762·43-s − 1.19·45-s + 1.60·47-s − 6/7·49-s + 0.560·51-s + 1.92·53-s + 1.61·55-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.568978317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.568978317\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 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
−14.94877344239925, −14.23068890161208, −14.00294609469677, −13.63131402845650, −13.06974702752837, −12.35765398675703, −11.72057703717561, −11.46624152551856, −10.65623763086407, −10.25104187839021, −9.854581770120246, −9.016108963293389, −8.594475767218412, −8.342575106400666, −7.067701747173553, −6.528705000731765, −6.180951911749364, −5.779829089916770, −5.024234459913193, −4.555689537975173, −3.708877130974218, −2.648849376367072, −2.316578521846599, −1.333982332192473, −0.8210662579039611,
0.8210662579039611, 1.333982332192473, 2.316578521846599, 2.648849376367072, 3.708877130974218, 4.555689537975173, 5.024234459913193, 5.779829089916770, 6.180951911749364, 6.528705000731765, 7.067701747173553, 8.342575106400666, 8.594475767218412, 9.016108963293389, 9.854581770120246, 10.25104187839021, 10.65623763086407, 11.46624152551856, 11.72057703717561, 12.35765398675703, 13.06974702752837, 13.63131402845650, 14.00294609469677, 14.23068890161208, 14.94877344239925
