| L(s) = 1 | + 3-s + 5-s − 7-s + 9-s − 6·11-s + 4·13-s + 15-s + 6·17-s − 6·19-s − 21-s + 4·23-s + 25-s + 27-s − 2·29-s + 2·31-s − 6·33-s − 35-s + 6·37-s + 4·39-s + 6·41-s + 12·43-s + 45-s + 49-s + 6·51-s − 4·53-s − 6·55-s − 6·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.80·11-s + 1.10·13-s + 0.258·15-s + 1.45·17-s − 1.37·19-s − 0.218·21-s + 0.834·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.359·31-s − 1.04·33-s − 0.169·35-s + 0.986·37-s + 0.640·39-s + 0.937·41-s + 1.82·43-s + 0.149·45-s + 1/7·49-s + 0.840·51-s − 0.549·53-s − 0.809·55-s − 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.368825692\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.368825692\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−8.514491413425512987354203964262, −7.956246768509965011958567646667, −7.31854398015984285823914960074, −6.22671753989414130759715404600, −5.71131059741604191464164554228, −4.80883390968464593547938291868, −3.79794282110674416515540780488, −2.91822828476048195678268450008, −2.25500705486060836822807831094, −0.899613123290815941295275561282,
0.899613123290815941295275561282, 2.25500705486060836822807831094, 2.91822828476048195678268450008, 3.79794282110674416515540780488, 4.80883390968464593547938291868, 5.71131059741604191464164554228, 6.22671753989414130759715404600, 7.31854398015984285823914960074, 7.956246768509965011958567646667, 8.514491413425512987354203964262
