L(s) = 1 | + 4·5-s + 6·7-s − 52·11-s − 26·13-s − 188·17-s − 74·19-s + 148·23-s − 58·25-s + 288·29-s + 248·31-s + 24·35-s − 342·37-s − 256·43-s + 132·47-s + 61·49-s + 952·53-s − 208·55-s + 1.00e3·59-s + 34·61-s − 104·65-s − 866·67-s + 776·71-s + 1.87e3·73-s − 312·77-s − 182·79-s + 1.33e3·83-s − 752·85-s + ⋯ |
L(s) = 1 | + 0.357·5-s + 0.323·7-s − 1.42·11-s − 0.554·13-s − 2.68·17-s − 0.893·19-s + 1.34·23-s − 0.463·25-s + 1.84·29-s + 1.43·31-s + 0.115·35-s − 1.51·37-s − 0.907·43-s + 0.409·47-s + 0.177·49-s + 2.46·53-s − 0.509·55-s + 2.21·59-s + 0.0713·61-s − 0.198·65-s − 1.57·67-s + 1.29·71-s + 3.00·73-s − 0.461·77-s − 0.259·79-s + 1.76·83-s − 0.959·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.853949408\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.853949408\) |
\(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
good | 5 | $D_{4}$ | \( 1 - 4 T + 74 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 6 T - 25 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 52 T + 1718 T^{2} + 52 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 p T + 3843 T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 188 T + 18482 T^{2} + 188 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 74 T + 3567 T^{2} + 74 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 148 T + 25310 T^{2} - 148 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 288 T + 57994 T^{2} - 288 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 p T + 63438 T^{2} - 8 p^{4} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 342 T + 112547 T^{2} + 342 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 46478 T^{2} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 256 T + 172518 T^{2} + 256 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 132 T + 211822 T^{2} - 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 952 T + 489050 T^{2} - 952 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 1004 T + 583382 T^{2} - 1004 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 34 T + 246171 T^{2} - 34 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 866 T + 742935 T^{2} + 866 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 776 T + 704366 T^{2} - 776 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 1874 T + 1630083 T^{2} - 1874 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 182 T + 872679 T^{2} + 182 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1336 T + 1208918 T^{2} - 1336 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 876 T + 1522402 T^{2} + 876 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 38 T + 431787 T^{2} + 38 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.961052865197363641262310979800, −8.570124414095437227670841300155, −8.560396172435585801143063160824, −8.159564077943009293287302148770, −7.56766164430615468217173622705, −6.93095979027148754673578902114, −6.83902802414491041001409447302, −6.54122648357259722620947717799, −5.97572279147423838195351248653, −5.30842806833994825936270834523, −4.92078423012640021035229024772, −4.91920545884727295483239446811, −4.15776798662165574568703946758, −3.91128900112937057346789878178, −2.82797732068924816986738738900, −2.76479877754158575867188921355, −2.04416846613523531582556795924, −1.99976455368404887097164336477, −0.70392372668962425286367885466, −0.51048522367466234792070860293,
0.51048522367466234792070860293, 0.70392372668962425286367885466, 1.99976455368404887097164336477, 2.04416846613523531582556795924, 2.76479877754158575867188921355, 2.82797732068924816986738738900, 3.91128900112937057346789878178, 4.15776798662165574568703946758, 4.91920545884727295483239446811, 4.92078423012640021035229024772, 5.30842806833994825936270834523, 5.97572279147423838195351248653, 6.54122648357259722620947717799, 6.83902802414491041001409447302, 6.93095979027148754673578902114, 7.56766164430615468217173622705, 8.159564077943009293287302148770, 8.560396172435585801143063160824, 8.570124414095437227670841300155, 8.961052865197363641262310979800