| L(s) = 1 | − 6·3-s − 12·7-s + 27·9-s + 24·11-s − 8·13-s − 56·17-s + 60·19-s + 72·21-s − 84·23-s − 108·27-s + 12·29-s + 132·31-s − 144·33-s + 104·37-s + 48·39-s + 140·41-s − 504·43-s − 348·47-s − 222·49-s + 336·51-s + 684·53-s − 360·57-s + 888·59-s − 172·61-s − 324·63-s − 1.56e3·67-s + 504·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.647·7-s + 9-s + 0.657·11-s − 0.170·13-s − 0.798·17-s + 0.724·19-s + 0.748·21-s − 0.761·23-s − 0.769·27-s + 0.0768·29-s + 0.764·31-s − 0.759·33-s + 0.462·37-s + 0.197·39-s + 0.533·41-s − 1.78·43-s − 1.08·47-s − 0.647·49-s + 0.922·51-s + 1.77·53-s − 0.836·57-s + 1.95·59-s − 0.361·61-s − 0.647·63-s − 2.84·67-s + 0.879·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $D_{4}$ | \( 1 + 12 T + 366 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 24 T + 1382 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 8 T + 1206 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 56 T + 7406 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 60 T + 11414 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 84 T + 17198 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p^{3} T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 132 T + 63582 T^{2} - 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 104 T - 52986 T^{2} - 104 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 140 T + 129926 T^{2} - 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 252 T + p^{3} T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 348 T + 220478 T^{2} + 348 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 684 T + 401902 T^{2} - 684 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 888 T + 538118 T^{2} - 888 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 172 T + 140958 T^{2} + 172 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 1560 T + 1204230 T^{2} + 1560 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 144 T + 356462 T^{2} - 144 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 564 T + 742214 T^{2} - 564 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 84 T + 765342 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1512 T + 1436006 T^{2} + 1512 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 28 T + 372038 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 148 T + 1779558 T^{2} + 148 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.520962263588853306753717389892, −8.068507573078765511216650948163, −7.33879371745502312764795854545, −7.33065547323719659962262486598, −6.72553433869754046692492426593, −6.48807066595065422805471813265, −6.09890424864793560631274424255, −5.82259638348506835656005190282, −5.20418968116134638260768869934, −4.99940705070347767626465083145, −4.29019612160379377136694186648, −4.22834864103475079255686921809, −3.55196361523812039974656481695, −3.13697734446375418732813453647, −2.52287856302304773002990517820, −1.97441713312742470373629187319, −1.31370929101134405974443965374, −0.930250843796304612711242119615, 0, 0,
0.930250843796304612711242119615, 1.31370929101134405974443965374, 1.97441713312742470373629187319, 2.52287856302304773002990517820, 3.13697734446375418732813453647, 3.55196361523812039974656481695, 4.22834864103475079255686921809, 4.29019612160379377136694186648, 4.99940705070347767626465083145, 5.20418968116134638260768869934, 5.82259638348506835656005190282, 6.09890424864793560631274424255, 6.48807066595065422805471813265, 6.72553433869754046692492426593, 7.33065547323719659962262486598, 7.33879371745502312764795854545, 8.068507573078765511216650948163, 8.520962263588853306753717389892
