| L(s) = 1 | + 6·2-s + 8·4-s + 24·5-s − 24·8-s − 54·9-s + 144·10-s + 296·13-s − 48·16-s − 600·17-s − 324·18-s + 192·20-s − 476·25-s + 1.77e3·26-s + 888·29-s − 384·32-s − 3.60e3·34-s − 432·36-s − 4.40e3·37-s − 576·40-s + 552·41-s − 1.29e3·45-s + 4.51e3·49-s − 2.85e3·50-s + 2.36e3·52-s + 5.11e3·53-s + 5.32e3·58-s + 4.23e3·61-s + ⋯ |
| L(s) = 1 | + 3/2·2-s + 1/2·4-s + 0.959·5-s − 3/8·8-s − 2/3·9-s + 1.43·10-s + 1.75·13-s − 0.187·16-s − 2.07·17-s − 18-s + 0.479·20-s − 0.761·25-s + 2.62·26-s + 1.05·29-s − 3/8·32-s − 3.11·34-s − 1/3·36-s − 3.21·37-s − 0.359·40-s + 0.328·41-s − 0.639·45-s + 1.88·49-s − 1.14·50-s + 0.875·52-s + 1.81·53-s + 1.58·58-s + 1.13·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.540242329\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.540242329\) |
| \(L(3)\) |
|
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 | $C_2^2$ | \( 1 - 3 p T + 7 p^{2} T^{2} - 3 p^{5} T^{3} + p^{8} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| good | 5 | $D_{4}$ | \( ( 1 - 12 T + 454 T^{2} - 12 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 4516 T^{2} + 16148934 T^{4} - 4516 p^{8} T^{6} + p^{16} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 36196 T^{2} + 708332166 T^{4} - 36196 p^{8} T^{6} + p^{16} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 148 T + 32646 T^{2} - 148 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 + 300 T + 186214 T^{2} + 300 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 195556 T^{2} + 17286835974 T^{4} - 195556 p^{8} T^{6} + p^{16} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 655492 T^{2} + 482373414 p^{2} T^{4} - 655492 p^{8} T^{6} + p^{16} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 444 T + 1423078 T^{2} - 444 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 67420 p T^{2} + 2465294161734 T^{4} - 67420 p^{9} T^{6} + p^{16} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 2204 T + 4842918 T^{2} + 2204 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 276 T + 5507494 T^{2} - 276 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 3593572 T^{2} + 5893643026566 T^{4} - 3593572 p^{8} T^{6} + p^{16} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 16853380 T^{2} + 118578854811654 T^{4} - 16853380 p^{8} T^{6} + p^{16} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 2556 T + 17173798 T^{2} - 2556 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 32722276 T^{2} + 559903452302214 T^{4} - 32722276 p^{8} T^{6} + p^{16} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 2116 T + 16830246 T^{2} - 2116 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 53256676 T^{2} + 1338152481850374 T^{4} - 53256676 p^{8} T^{6} + p^{16} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 79127428 T^{2} + 2740885222137990 T^{4} - 79127428 p^{8} T^{6} + p^{16} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 4420 T + 3693510 T^{2} - 4420 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 86136100 T^{2} + 4459690111983174 T^{4} - 86136100 p^{8} T^{6} + p^{16} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 855268 T^{2} + 4298268871421190 T^{4} - 855268 p^{8} T^{6} + p^{16} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 12540 T + 132835270 T^{2} + 12540 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 11524 T + 146880774 T^{2} - 11524 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.39756887227521205062159558897, −14.25169970326605343776020203273, −13.85046118940692215806213530802, −13.69443596423998722939989779762, −13.26630440922724312702570987254, −13.24138420205178385669881740226, −12.64736754533774355715483861521, −12.00448445958351114897360335675, −11.93045416105845690846734904875, −11.15466798320034113788122280773, −10.96109952932730072767385126388, −10.31970931103209328325156031806, −10.16515386371985889655554108940, −9.191546723912840722951423572117, −8.804601982129558388563576446641, −8.714537531290301440109738135882, −8.041765237760488827793366948790, −6.86371007993304063065754336602, −6.77211105201542335933981436986, −5.75172618691198996991299154693, −5.74892451087126920716738699033, −4.94391572785138053514189951270, −4.11501248978157856650536640493, −3.57829799171819653973573853193, −2.19838578255000481774688793477,
2.19838578255000481774688793477, 3.57829799171819653973573853193, 4.11501248978157856650536640493, 4.94391572785138053514189951270, 5.74892451087126920716738699033, 5.75172618691198996991299154693, 6.77211105201542335933981436986, 6.86371007993304063065754336602, 8.041765237760488827793366948790, 8.714537531290301440109738135882, 8.804601982129558388563576446641, 9.191546723912840722951423572117, 10.16515386371985889655554108940, 10.31970931103209328325156031806, 10.96109952932730072767385126388, 11.15466798320034113788122280773, 11.93045416105845690846734904875, 12.00448445958351114897360335675, 12.64736754533774355715483861521, 13.24138420205178385669881740226, 13.26630440922724312702570987254, 13.69443596423998722939989779762, 13.85046118940692215806213530802, 14.25169970326605343776020203273, 14.39756887227521205062159558897
