| L(s) = 1 | + 62·5-s + 98·7-s − 972·11-s + 78·13-s − 560·17-s + 2.64e3·19-s − 2.27e3·23-s + 1.05e3·25-s + 7.80e3·29-s + 5.44e3·31-s + 6.07e3·35-s + 576·37-s + 1.68e4·41-s − 8.39e3·43-s + 4.53e3·47-s + 7.20e3·49-s − 1.42e3·53-s − 6.02e4·55-s − 3.41e4·59-s + 1.91e4·61-s + 4.83e3·65-s + 5.69e4·67-s + 7.22e3·71-s + 1.28e5·73-s − 9.52e4·77-s + 5.28e4·79-s + 8.44e4·83-s + ⋯ |
| L(s) = 1 | + 1.10·5-s + 0.755·7-s − 2.42·11-s + 0.128·13-s − 0.469·17-s + 1.67·19-s − 0.895·23-s + 0.338·25-s + 1.72·29-s + 1.01·31-s + 0.838·35-s + 0.0691·37-s + 1.56·41-s − 0.692·43-s + 0.299·47-s + 3/7·49-s − 0.0694·53-s − 2.68·55-s − 1.27·59-s + 0.657·61-s + 0.141·65-s + 1.54·67-s + 0.170·71-s + 2.82·73-s − 1.83·77-s + 0.951·79-s + 1.34·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.967102258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.967102258\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 62 T + 2786 T^{2} - 62 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 972 T + 551926 T^{2} + 972 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 6 p T - 148150 T^{2} - 6 p^{6} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 560 T + 2911742 T^{2} + 560 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2642 T + 6454926 T^{2} - 2642 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2272 T + 3276974 T^{2} + 2272 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 7808 T + 56228822 T^{2} - 7808 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 5444 T + 61676286 T^{2} - 5444 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 576 T + 63988358 T^{2} - 576 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 16888 T + 277723070 T^{2} - 16888 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8396 T + 291418902 T^{2} + 8396 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4532 T + 192546782 T^{2} - 4532 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 1420 T + 601997678 T^{2} + 1420 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 34146 T + 1702640302 T^{2} + 34146 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 19106 T - 202373982 T^{2} - 19106 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 56952 T + 3238977590 T^{2} - 56952 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 7224 T + 3534775246 T^{2} - 7224 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 128828 T + 8014301382 T^{2} - 128828 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 52808 T + 6427138206 T^{2} - 52808 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 84486 T + 4859890798 T^{2} - 84486 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 130972 T + 14510409206 T^{2} - 130972 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 194624 T + 26622623358 T^{2} - 194624 p^{5} T^{3} + p^{10} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.36642534538784457901632765727, −9.898991370242585629651535132176, −9.591656256696332292364270270137, −9.138289492770311183793425115355, −8.388355494805211027420834492546, −8.012821283629397858902762720153, −7.83270108465493667520821741688, −7.33852326907518346972394024383, −6.38970651874797197063000701809, −6.36233767872178641266434230549, −5.47788046569421298790724544548, −5.24261944571308483278222942633, −4.90453838959611915338473952714, −4.32142900050270701478367841349, −3.37962327418072056594856622378, −2.84273132491704866433304713472, −2.20794676708162849652633382074, −2.05373339389832616561133157460, −0.897039847065639093826117081168, −0.63194354444668070666829272954,
0.63194354444668070666829272954, 0.897039847065639093826117081168, 2.05373339389832616561133157460, 2.20794676708162849652633382074, 2.84273132491704866433304713472, 3.37962327418072056594856622378, 4.32142900050270701478367841349, 4.90453838959611915338473952714, 5.24261944571308483278222942633, 5.47788046569421298790724544548, 6.36233767872178641266434230549, 6.38970651874797197063000701809, 7.33852326907518346972394024383, 7.83270108465493667520821741688, 8.012821283629397858902762720153, 8.388355494805211027420834492546, 9.138289492770311183793425115355, 9.591656256696332292364270270137, 9.898991370242585629651535132176, 10.36642534538784457901632765727
