| L(s) = 1 | − 5-s + 6·7-s − 7·11-s − 2·13-s + 3·17-s + 5·19-s + 3·23-s − 6·29-s + 4·31-s − 6·35-s + 4·37-s + 6·41-s − 43-s − 5·47-s + 13·49-s + 7·53-s + 7·55-s + 8·59-s + 4·61-s + 2·65-s − 67-s + 18·71-s + 73-s − 42·77-s − 3·79-s + 10·83-s − 3·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 2.26·7-s − 2.11·11-s − 0.554·13-s + 0.727·17-s + 1.14·19-s + 0.625·23-s − 1.11·29-s + 0.718·31-s − 1.01·35-s + 0.657·37-s + 0.937·41-s − 0.152·43-s − 0.729·47-s + 13/7·49-s + 0.961·53-s + 0.943·55-s + 1.04·59-s + 0.512·61-s + 0.248·65-s − 0.122·67-s + 2.13·71-s + 0.117·73-s − 4.78·77-s − 0.337·79-s + 1.09·83-s − 0.325·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7884864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7884864 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.876867551\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.876867551\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 + 7 T + 25 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 27 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 5 T + 35 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3 T + 39 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 41 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 5 T + 91 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 7 T + 35 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 97 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 89 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + T + 125 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 18 T + 186 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - T + 137 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 3 T + 151 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 10 T + 43 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 10 T + 166 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.879362355890751388495600498052, −8.344241320008426142245165340165, −8.096410662848075322484707743849, −7.88060036145627466410727690064, −7.53533086352103418875921917142, −7.51090592570623155238764603416, −6.78738342729483370090340437160, −6.40923597379874006888518342035, −5.44579349769948778778167434380, −5.37675770511343716005985018045, −5.26602315022340087374891366179, −4.89679389332726324543797740120, −4.16842007631476620220672457726, −4.11368578530137730870355019056, −3.12515788144944815217133519179, −2.96740349544672909585815028284, −2.21776487149155031562013161237, −1.98271913699416622622542289164, −1.14438638270624478872704677405, −0.60808052056108864109542004150,
0.60808052056108864109542004150, 1.14438638270624478872704677405, 1.98271913699416622622542289164, 2.21776487149155031562013161237, 2.96740349544672909585815028284, 3.12515788144944815217133519179, 4.11368578530137730870355019056, 4.16842007631476620220672457726, 4.89679389332726324543797740120, 5.26602315022340087374891366179, 5.37675770511343716005985018045, 5.44579349769948778778167434380, 6.40923597379874006888518342035, 6.78738342729483370090340437160, 7.51090592570623155238764603416, 7.53533086352103418875921917142, 7.88060036145627466410727690064, 8.096410662848075322484707743849, 8.344241320008426142245165340165, 8.879362355890751388495600498052
