| L(s) = 1 | − 1.70·3-s − 5-s − 1.07·7-s − 0.0783·9-s + 6.04·11-s + 0.290·13-s + 1.70·15-s + 1.07·17-s + 5.26·19-s + 1.84·21-s + 4.34·23-s + 25-s + 5.26·27-s + 9.31·29-s − 31-s − 10.3·33-s + 1.07·35-s + 2.44·37-s − 0.496·39-s + 5.60·41-s − 7.86·43-s + 0.0783·45-s + 1.75·47-s − 5.83·49-s − 1.84·51-s + 6.44·53-s − 6.04·55-s + ⋯ |
| L(s) = 1 | − 0.986·3-s − 0.447·5-s − 0.407·7-s − 0.0261·9-s + 1.82·11-s + 0.0806·13-s + 0.441·15-s + 0.261·17-s + 1.20·19-s + 0.402·21-s + 0.904·23-s + 0.200·25-s + 1.01·27-s + 1.72·29-s − 0.179·31-s − 1.79·33-s + 0.182·35-s + 0.402·37-s − 0.0795·39-s + 0.874·41-s − 1.19·43-s + 0.0116·45-s + 0.256·47-s − 0.833·49-s − 0.258·51-s + 0.885·53-s − 0.815·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.540005937\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.540005937\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 3 | \( 1 + 1.70T + 3T^{2} \) |
| 7 | \( 1 + 1.07T + 7T^{2} \) |
| 11 | \( 1 - 6.04T + 11T^{2} \) |
| 13 | \( 1 - 0.290T + 13T^{2} \) |
| 17 | \( 1 - 1.07T + 17T^{2} \) |
| 19 | \( 1 - 5.26T + 19T^{2} \) |
| 23 | \( 1 - 4.34T + 23T^{2} \) |
| 29 | \( 1 - 9.31T + 29T^{2} \) |
| 37 | \( 1 - 2.44T + 37T^{2} \) |
| 41 | \( 1 - 5.60T + 41T^{2} \) |
| 43 | \( 1 + 7.86T + 43T^{2} \) |
| 47 | \( 1 - 1.75T + 47T^{2} \) |
| 53 | \( 1 - 6.44T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 - 12.0T + 61T^{2} \) |
| 67 | \( 1 + 13.9T + 67T^{2} \) |
| 71 | \( 1 - 4.68T + 71T^{2} \) |
| 73 | \( 1 + 8.18T + 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 - 3.55T + 83T^{2} \) |
| 89 | \( 1 - 3.84T + 89T^{2} \) |
| 97 | \( 1 - 2.49T + 97T^{2} \) |
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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
−7.40641222997200630122098575776, −6.83544760383778025041872334052, −6.36888817732764031428785057224, −5.67850489451272241707763112877, −4.96213953207443853711064384023, −4.24898519744591626945875110203, −3.46392155237107508645754946568, −2.78671536354882387676334270369, −1.28402174102916921542625869391, −0.72669073908532684573046151445,
0.72669073908532684573046151445, 1.28402174102916921542625869391, 2.78671536354882387676334270369, 3.46392155237107508645754946568, 4.24898519744591626945875110203, 4.96213953207443853711064384023, 5.67850489451272241707763112877, 6.36888817732764031428785057224, 6.83544760383778025041872334052, 7.40641222997200630122098575776
