| L(s) = 1 | + 2-s + 4-s − 3.72·5-s − 2.68·7-s + 8-s − 3.72·10-s − 3.03·11-s − 1.51·13-s − 2.68·14-s + 16-s − 3.16·17-s − 7.34·19-s − 3.72·20-s − 3.03·22-s − 4.36·23-s + 8.89·25-s − 1.51·26-s − 2.68·28-s + 3.00·29-s − 1.76·31-s + 32-s − 3.16·34-s + 10.0·35-s − 11.0·37-s − 7.34·38-s − 3.72·40-s + 7.36·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.66·5-s − 1.01·7-s + 0.353·8-s − 1.17·10-s − 0.915·11-s − 0.420·13-s − 0.717·14-s + 0.250·16-s − 0.767·17-s − 1.68·19-s − 0.833·20-s − 0.647·22-s − 0.909·23-s + 1.77·25-s − 0.297·26-s − 0.507·28-s + 0.557·29-s − 0.316·31-s + 0.176·32-s − 0.542·34-s + 1.69·35-s − 1.80·37-s − 1.19·38-s − 0.589·40-s + 1.14·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4355550913\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4355550913\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 5 | \( 1 + 3.72T + 5T^{2} \) |
| 7 | \( 1 + 2.68T + 7T^{2} \) |
| 11 | \( 1 + 3.03T + 11T^{2} \) |
| 13 | \( 1 + 1.51T + 13T^{2} \) |
| 17 | \( 1 + 3.16T + 17T^{2} \) |
| 19 | \( 1 + 7.34T + 19T^{2} \) |
| 23 | \( 1 + 4.36T + 23T^{2} \) |
| 29 | \( 1 - 3.00T + 29T^{2} \) |
| 31 | \( 1 + 1.76T + 31T^{2} \) |
| 37 | \( 1 + 11.0T + 37T^{2} \) |
| 41 | \( 1 - 7.36T + 41T^{2} \) |
| 43 | \( 1 + 4.31T + 43T^{2} \) |
| 47 | \( 1 + 4.55T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 - 3.61T + 59T^{2} \) |
| 61 | \( 1 + 1.93T + 61T^{2} \) |
| 67 | \( 1 - 0.348T + 67T^{2} \) |
| 71 | \( 1 - 1.77T + 71T^{2} \) |
| 73 | \( 1 + 9.05T + 73T^{2} \) |
| 79 | \( 1 - 8.59T + 79T^{2} \) |
| 83 | \( 1 + 0.134T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.76365280489287020720613544058, −6.98088491209953580775726139565, −6.61459968583467145045170948141, −5.71966777660850917647923954277, −4.80997399469067168630278871388, −4.21259849600942967464730245536, −3.64982870847409689764283640042, −2.88565278645245140851480239727, −2.09945992316997258607393051331, −0.26877334130636803071846390598,
0.26877334130636803071846390598, 2.09945992316997258607393051331, 2.88565278645245140851480239727, 3.64982870847409689764283640042, 4.21259849600942967464730245536, 4.80997399469067168630278871388, 5.71966777660850917647923954277, 6.61459968583467145045170948141, 6.98088491209953580775726139565, 7.76365280489287020720613544058
