| L(s) = 1 | + 19·2-s + 99·4-s + 353·5-s − 2.00e3·7-s − 665·8-s + 6.70e3·10-s + 1.81e3·11-s + 4.39e3·13-s − 3.81e4·14-s − 8.74e3·16-s + 2.53e4·17-s + 2.21e4·19-s + 3.49e4·20-s + 3.43e4·22-s + 2.64e4·23-s − 5.25e4·25-s + 8.34e4·26-s − 1.98e5·28-s + 5.80e3·29-s + 3.97e4·31-s − 6.25e4·32-s + 4.81e5·34-s − 7.09e5·35-s + 1.63e5·37-s + 4.20e5·38-s − 2.34e5·40-s + 3.30e5·41-s + ⋯ |
| L(s) = 1 | + 1.67·2-s + 0.773·4-s + 1.26·5-s − 2.21·7-s − 0.459·8-s + 2.12·10-s + 0.410·11-s + 0.554·13-s − 3.71·14-s − 0.533·16-s + 1.25·17-s + 0.739·19-s + 0.976·20-s + 0.688·22-s + 0.452·23-s − 0.673·25-s + 0.931·26-s − 1.71·28-s + 0.0441·29-s + 0.239·31-s − 0.337·32-s + 2.10·34-s − 2.79·35-s + 0.530·37-s + 1.24·38-s − 0.579·40-s + 0.749·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(6.070903529\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.070903529\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - 19 T + 131 p T^{2} - 19 p^{7} T^{3} + p^{14} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 353 T + 177208 T^{2} - 353 p^{7} T^{3} + p^{14} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 41 p^{2} T + 2649282 T^{2} + 41 p^{9} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 1810 T + 7198390 T^{2} - 1810 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 25361 T + 643460668 T^{2} - 25361 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 22106 T + 141106790 T^{2} - 22106 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 26424 T + 3097648846 T^{2} - 26424 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 5804 T + 33449999614 T^{2} - 5804 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 39744 T + 54390419758 T^{2} - 39744 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 163299 T + 181994892160 T^{2} - 163299 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 8070 p T + 274020654610 T^{2} - 8070 p^{8} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 229307 T + 555183291698 T^{2} - 229307 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 1638525 T + 1682936215978 T^{2} - 1638525 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 1046382 T + 1701182112730 T^{2} + 1046382 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 370158 T + 4837245961366 T^{2} - 370158 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4675422 T + 11232926947738 T^{2} - 4675422 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 1821402 T + 6715905784390 T^{2} + 1821402 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1135611 T - 6597281929502 T^{2} - 1135611 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 6459284 T + 32251656236358 T^{2} + 6459284 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 73808 T + 15935358386334 T^{2} + 73808 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12100972 T + 88634017455958 T^{2} - 12100972 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 9815060 T + 78068771423926 T^{2} + 9815060 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 17591688 T + 206995518168430 T^{2} + 17591688 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−12.67700319885120628685264419364, −12.26965027724111893266582371024, −11.72485089122157560969847532469, −10.91027889791883448433120279748, −10.06675721833583946532621680108, −9.910978636721461569128142735574, −9.284466192023272657526554764589, −9.103527094935488790446634898424, −7.977143208677111252535007637960, −7.20173200335642944240620971555, −6.47456536160393936660409162111, −6.09327224043383856987320963124, −5.55418711534165943792088652573, −5.34670468925610430090815578629, −4.01997576282965516641163069967, −3.90113043819529160513021911782, −3.03150718480504129871974933456, −2.67072641756457666423542882651, −1.41937304622289051252595034820, −0.56525424238042424750089976177,
0.56525424238042424750089976177, 1.41937304622289051252595034820, 2.67072641756457666423542882651, 3.03150718480504129871974933456, 3.90113043819529160513021911782, 4.01997576282965516641163069967, 5.34670468925610430090815578629, 5.55418711534165943792088652573, 6.09327224043383856987320963124, 6.47456536160393936660409162111, 7.20173200335642944240620971555, 7.977143208677111252535007637960, 9.103527094935488790446634898424, 9.284466192023272657526554764589, 9.910978636721461569128142735574, 10.06675721833583946532621680108, 10.91027889791883448433120279748, 11.72485089122157560969847532469, 12.26965027724111893266582371024, 12.67700319885120628685264419364
