L(s) = 1 | + 2-s + 4-s + 8-s + 4·13-s + 16-s + 6·17-s − 2·19-s − 5·25-s + 4·26-s + 6·29-s + 4·31-s + 32-s + 6·34-s + 2·37-s − 2·38-s + 6·41-s + 8·43-s − 12·47-s − 5·50-s + 4·52-s − 6·53-s + 6·58-s − 6·59-s − 8·61-s + 4·62-s + 64-s − 4·67-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 1.10·13-s + 1/4·16-s + 1.45·17-s − 0.458·19-s − 25-s + 0.784·26-s + 1.11·29-s + 0.718·31-s + 0.176·32-s + 1.02·34-s + 0.328·37-s − 0.324·38-s + 0.937·41-s + 1.21·43-s − 1.75·47-s − 0.707·50-s + 0.554·52-s − 0.824·53-s + 0.787·58-s − 0.781·59-s − 1.02·61-s + 0.508·62-s + 1/8·64-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.594185568\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.594185568\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−10.27339907123230780461643050534, −9.393998538973797562285424645673, −8.245010328271296459357780366393, −7.64010214491199898083268392854, −6.37349457460642388728354325328, −5.88407913223931048395780461484, −4.74683922157615295493069282209, −3.79391663707115759548696099652, −2.83833445427180997893704571146, −1.33827299261372059142982522511,
1.33827299261372059142982522511, 2.83833445427180997893704571146, 3.79391663707115759548696099652, 4.74683922157615295493069282209, 5.88407913223931048395780461484, 6.37349457460642388728354325328, 7.64010214491199898083268392854, 8.245010328271296459357780366393, 9.393998538973797562285424645673, 10.27339907123230780461643050534