| L(s) = 1 | + 2-s + 4-s − 1.73·5-s + 8-s − 1.73·10-s + 6.92·13-s + 16-s − 6·17-s − 1.73·19-s − 1.73·20-s − 3.46·23-s − 2.00·25-s + 6.92·26-s − 9·29-s + 4·31-s + 32-s − 6·34-s − 2·37-s − 1.73·38-s − 1.73·40-s + 5.19·43-s − 3.46·46-s − 10.3·47-s − 7·49-s − 2.00·50-s + 6.92·52-s − 9·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.774·5-s + 0.353·8-s − 0.547·10-s + 1.92·13-s + 0.250·16-s − 1.45·17-s − 0.397·19-s − 0.387·20-s − 0.722·23-s − 0.400·25-s + 1.35·26-s − 1.67·29-s + 0.718·31-s + 0.176·32-s − 1.02·34-s − 0.328·37-s − 0.280·38-s − 0.273·40-s + 0.792·43-s − 0.510·46-s − 1.51·47-s − 49-s − 0.282·50-s + 0.960·52-s − 1.18·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 1.73T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 13 | \( 1 - 6.92T + 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + 1.73T + 19T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 + 9T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 5.19T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 3.46T + 59T^{2} \) |
| 61 | \( 1 + 3.46T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 - 5.19T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 - 3.46T + 89T^{2} \) |
| 97 | \( 1 + 7T + 97T^{2} \) |
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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
−7.77742066731716101047692693734, −6.62621828061047586990784655440, −6.38095463536672613753317149386, −5.53016257617462970631437634612, −4.62049184555144946356221089625, −3.84309389370643789778772744051, −3.61087265270655574326986980101, −2.37166067365396176853733959563, −1.47988821772183595374843031263, 0,
1.47988821772183595374843031263, 2.37166067365396176853733959563, 3.61087265270655574326986980101, 3.84309389370643789778772744051, 4.62049184555144946356221089625, 5.53016257617462970631437634612, 6.38095463536672613753317149386, 6.62621828061047586990784655440, 7.77742066731716101047692693734
