| L(s) = 1 | + 2-s + 4-s + 1.26·5-s + 2.94·7-s + 8-s + 1.26·10-s − 3.29·11-s − 3.36·13-s + 2.94·14-s + 16-s + 7.44·19-s + 1.26·20-s − 3.29·22-s + 1.41·23-s − 3.40·25-s − 3.36·26-s + 2.94·28-s − 0.522·29-s + 7.10·31-s + 32-s + 3.71·35-s − 0.792·37-s + 7.44·38-s + 1.26·40-s − 4.06·41-s + 0.867·43-s − 3.29·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.564·5-s + 1.11·7-s + 0.353·8-s + 0.399·10-s − 0.994·11-s − 0.932·13-s + 0.787·14-s + 0.250·16-s + 1.70·19-s + 0.282·20-s − 0.703·22-s + 0.294·23-s − 0.681·25-s − 0.659·26-s + 0.556·28-s − 0.0970·29-s + 1.27·31-s + 0.176·32-s + 0.628·35-s − 0.130·37-s + 1.20·38-s + 0.199·40-s − 0.634·41-s + 0.132·43-s − 0.497·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5202 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5202 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.966109347\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.966109347\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 - 1.26T + 5T^{2} \) |
| 7 | \( 1 - 2.94T + 7T^{2} \) |
| 11 | \( 1 + 3.29T + 11T^{2} \) |
| 13 | \( 1 + 3.36T + 13T^{2} \) |
| 19 | \( 1 - 7.44T + 19T^{2} \) |
| 23 | \( 1 - 1.41T + 23T^{2} \) |
| 29 | \( 1 + 0.522T + 29T^{2} \) |
| 31 | \( 1 - 7.10T + 31T^{2} \) |
| 37 | \( 1 + 0.792T + 37T^{2} \) |
| 41 | \( 1 + 4.06T + 41T^{2} \) |
| 43 | \( 1 - 0.867T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 - 9.98T + 53T^{2} \) |
| 59 | \( 1 - 7.31T + 59T^{2} \) |
| 61 | \( 1 - 10.7T + 61T^{2} \) |
| 67 | \( 1 - 13.5T + 67T^{2} \) |
| 71 | \( 1 + 7.38T + 71T^{2} \) |
| 73 | \( 1 + 5.23T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 + 12.3T + 83T^{2} \) |
| 89 | \( 1 - 17.3T + 89T^{2} \) |
| 97 | \( 1 + 6.22T + 97T^{2} \) |
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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
−7.974671089101284827295432185112, −7.49673284289050183936472865039, −6.81426948630046584214423587751, −5.67783463355262022614428637183, −5.28401908154250672286948797444, −4.77280600118464391150608249004, −3.78774790761151379129118752916, −2.69767319003587140897800181017, −2.16852879630294157004684214224, −1.00710629815589260071665235009,
1.00710629815589260071665235009, 2.16852879630294157004684214224, 2.69767319003587140897800181017, 3.78774790761151379129118752916, 4.77280600118464391150608249004, 5.28401908154250672286948797444, 5.67783463355262022614428637183, 6.81426948630046584214423587751, 7.49673284289050183936472865039, 7.974671089101284827295432185112
