| L(s) = 1 | + 2-s + 4-s + 5-s + 3·7-s + 8-s + 10-s − 3·11-s + 3·14-s + 16-s + 17-s + 20-s − 3·22-s − 3·23-s + 25-s + 3·28-s − 6·29-s − 4·31-s + 32-s + 34-s + 3·35-s + 4·37-s + 40-s − 2·41-s − 43-s − 3·44-s − 3·46-s + 6·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 1.13·7-s + 0.353·8-s + 0.316·10-s − 0.904·11-s + 0.801·14-s + 1/4·16-s + 0.242·17-s + 0.223·20-s − 0.639·22-s − 0.625·23-s + 1/5·25-s + 0.566·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s + 0.171·34-s + 0.507·35-s + 0.657·37-s + 0.158·40-s − 0.312·41-s − 0.152·43-s − 0.452·44-s − 0.442·46-s + 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 258570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 258570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.05654780110249, −12.75650569730251, −12.16365947039359, −11.57711652140355, −11.32554895517637, −10.87988477822462, −10.27800552004812, −10.02299864743769, −9.400252528614252, −8.714171721814426, −8.401132763590374, −7.662534401891300, −7.553871549082204, −6.947904676098011, −6.245267823915779, −5.767635920209653, −5.312949236525312, −5.050530217622557, −4.396629786617378, −3.863687449178382, −3.380221945525082, −2.483381179438112, −2.264311100145717, −1.628161488559511, −0.9850932761583518, 0,
0.9850932761583518, 1.628161488559511, 2.264311100145717, 2.483381179438112, 3.380221945525082, 3.863687449178382, 4.396629786617378, 5.050530217622557, 5.312949236525312, 5.767635920209653, 6.245267823915779, 6.947904676098011, 7.553871549082204, 7.662534401891300, 8.401132763590374, 8.714171721814426, 9.400252528614252, 10.02299864743769, 10.27800552004812, 10.87988477822462, 11.32554895517637, 11.57711652140355, 12.16365947039359, 12.75650569730251, 13.05654780110249
