| L(s) = 1 | + 6·3-s − 3·5-s + 27·9-s + 51·11-s + 61·13-s − 18·15-s + 24·17-s − 169·19-s + 192·23-s − 115·25-s + 108·27-s − 39·29-s + 92·31-s + 306·33-s − 173·37-s + 366·39-s + 174·41-s + 497·43-s − 81·45-s − 180·47-s + 144·51-s + 285·53-s − 153·55-s − 1.01e3·57-s + 1.26e3·59-s + 328·61-s − 183·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.268·5-s + 9-s + 1.39·11-s + 1.30·13-s − 0.309·15-s + 0.342·17-s − 2.04·19-s + 1.74·23-s − 0.919·25-s + 0.769·27-s − 0.249·29-s + 0.533·31-s + 1.61·33-s − 0.768·37-s + 1.50·39-s + 0.662·41-s + 1.76·43-s − 0.268·45-s − 0.558·47-s + 0.395·51-s + 0.738·53-s − 0.375·55-s − 2.35·57-s + 2.80·59-s + 0.688·61-s − 0.349·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(9.334891853\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.334891853\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 + 3 T + 124 T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 51 T + 3184 T^{2} - 51 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 61 T + 2118 T^{2} - 61 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 24 T + 1762 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 169 T + 20730 T^{2} + 169 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 96 T + p^{3} T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 39 T + 12094 T^{2} + 39 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 92 T + 48873 T^{2} - 92 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 173 T + 62490 T^{2} + 173 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 174 T - 2846 T^{2} - 174 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 497 T + 174468 T^{2} - 497 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 180 T + 182914 T^{2} + 180 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 285 T + 224566 T^{2} - 285 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 1269 T + 13766 p T^{2} - 1269 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 328 T + 314646 T^{2} - 328 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 875 T + 782544 T^{2} - 875 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1404 T + 1077298 T^{2} - 1404 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1361 T + 1184556 T^{2} + 1361 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 182 T + 992307 T^{2} - 182 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 399 T + 946240 T^{2} - 399 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 822 T + 1516786 T^{2} - 822 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 841 T + 1945608 T^{2} - 841 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.874499298812786358990906812186, −8.447559572304077351751187363385, −8.115870360959972356113883825383, −7.938607024814203285967374030786, −7.15930808586014250450550888005, −6.97965011952350634833479983560, −6.45009285383483208726035701550, −6.44837626771959804191830795619, −5.53954422702416709482690543584, −5.49365689469031827991917415769, −4.59439271856733002744055599031, −4.23762738180793308815101525197, −3.78402675770221741379897159764, −3.77336641736451456791802480833, −3.13405290343870481724624930628, −2.52127845396631951456084097280, −2.07648689982123181363826727206, −1.64441408293455628944193370202, −0.78321511856135708691518449330, −0.77560998785981849426950730511,
0.77560998785981849426950730511, 0.78321511856135708691518449330, 1.64441408293455628944193370202, 2.07648689982123181363826727206, 2.52127845396631951456084097280, 3.13405290343870481724624930628, 3.77336641736451456791802480833, 3.78402675770221741379897159764, 4.23762738180793308815101525197, 4.59439271856733002744055599031, 5.49365689469031827991917415769, 5.53954422702416709482690543584, 6.44837626771959804191830795619, 6.45009285383483208726035701550, 6.97965011952350634833479983560, 7.15930808586014250450550888005, 7.938607024814203285967374030786, 8.115870360959972356113883825383, 8.447559572304077351751187363385, 8.874499298812786358990906812186
