| L(s) = 1 | − 2-s − 4-s + 5-s + 3·8-s − 10-s + 6·13-s − 16-s + 2·17-s + 8·19-s − 20-s − 8·23-s + 25-s − 6·26-s + 2·29-s − 4·31-s − 5·32-s − 2·34-s − 2·37-s − 8·38-s + 3·40-s − 6·41-s + 4·43-s + 8·46-s + 8·47-s − 50-s − 6·52-s − 10·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.447·5-s + 1.06·8-s − 0.316·10-s + 1.66·13-s − 1/4·16-s + 0.485·17-s + 1.83·19-s − 0.223·20-s − 1.66·23-s + 1/5·25-s − 1.17·26-s + 0.371·29-s − 0.718·31-s − 0.883·32-s − 0.342·34-s − 0.328·37-s − 1.29·38-s + 0.474·40-s − 0.937·41-s + 0.609·43-s + 1.17·46-s + 1.16·47-s − 0.141·50-s − 0.832·52-s − 1.37·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.279510442\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.279510442\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−9.158915116725801964521352538031, −8.275102727179600443330310060644, −7.82733728798443659692174656726, −6.82600380115164878858049675491, −5.81605182603986018580602603999, −5.25448844749250196408410433194, −4.05504421439464721652879145719, −3.34833988846054044432199186298, −1.79849192746842450415563444508, −0.888154170390839902230274080915,
0.888154170390839902230274080915, 1.79849192746842450415563444508, 3.34833988846054044432199186298, 4.05504421439464721652879145719, 5.25448844749250196408410433194, 5.81605182603986018580602603999, 6.82600380115164878858049675491, 7.82733728798443659692174656726, 8.275102727179600443330310060644, 9.158915116725801964521352538031
