| L(s) = 1 | − 6·3-s − 2·4-s − 6·5-s + 8·7-s + 9·9-s + 18·11-s + 12·12-s + 36·15-s − 30·17-s + 6·19-s + 12·20-s − 48·21-s + 30·23-s − 5·25-s + 18·27-s − 16·28-s + 48·29-s − 42·31-s − 108·33-s − 48·35-s − 18·36-s − 62·37-s − 8·43-s − 36·44-s − 54·45-s + 174·47-s + 22·49-s + ⋯ |
| L(s) = 1 | − 2·3-s − 1/2·4-s − 6/5·5-s + 8/7·7-s + 9-s + 1.63·11-s + 12-s + 12/5·15-s − 1.76·17-s + 6/19·19-s + 3/5·20-s − 2.28·21-s + 1.30·23-s − 1/5·25-s + 2/3·27-s − 4/7·28-s + 1.65·29-s − 1.35·31-s − 3.27·33-s − 1.37·35-s − 1/2·36-s − 1.67·37-s − 0.186·43-s − 0.818·44-s − 6/5·45-s + 3.70·47-s + 0.448·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1972708366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1972708366\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 8 T + 6 p T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 2 p T + p^{3} T^{2} + 10 p^{2} T^{3} + 28 p^{2} T^{4} + 10 p^{4} T^{5} + p^{7} T^{6} + 2 p^{7} T^{7} + p^{8} T^{8} \) |
| 5 | $C_2$$\times$$C_2^2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2}( 1 + 2 T - 21 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 11 | $D_4\times C_2$ | \( 1 - 18 T + 19 T^{2} - 1134 T^{3} + 39180 T^{4} - 1134 p^{2} T^{5} + 19 p^{4} T^{6} - 18 p^{6} T^{7} + p^{8} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 412 T^{2} + 89190 T^{4} - 412 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 30 T + 929 T^{2} + 1110 p T^{3} + 1380 p^{2} T^{4} + 1110 p^{3} T^{5} + 929 p^{4} T^{6} + 30 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 6 T + 731 T^{2} - 4314 T^{3} + 390972 T^{4} - 4314 p^{2} T^{5} + 731 p^{4} T^{6} - 6 p^{6} T^{7} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 30 T - 221 T^{2} - 1890 T^{3} + 500700 T^{4} - 1890 p^{2} T^{5} - 221 p^{4} T^{6} - 30 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 24 T + 1754 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 42 T + 1307 T^{2} + 30198 T^{3} + 158508 T^{4} + 30198 p^{2} T^{5} + 1307 p^{4} T^{6} + 42 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 62 T + 1297 T^{2} - 11842 T^{3} - 649388 T^{4} - 11842 p^{2} T^{5} + 1297 p^{4} T^{6} + 62 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 5500 T^{2} + 13185222 T^{4} - 5500 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 4 T + 3630 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 174 T + 17027 T^{2} - 1206690 T^{3} + 65507772 T^{4} - 1206690 p^{2} T^{5} + 17027 p^{4} T^{6} - 174 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 78 T - 767 T^{2} + 96174 T^{3} + 21955764 T^{4} + 96174 p^{2} T^{5} - 767 p^{4} T^{6} + 78 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 78 T + 5747 T^{2} + 290082 T^{3} + 8773068 T^{4} + 290082 p^{2} T^{5} + 5747 p^{4} T^{6} + 78 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 42 T + 2033 T^{2} + 60690 T^{3} - 9569868 T^{4} + 60690 p^{2} T^{5} + 2033 p^{4} T^{6} + 42 p^{6} T^{7} + p^{8} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 58 T - 2405 T^{2} - 186122 T^{3} - 1970756 T^{4} - 186122 p^{2} T^{5} - 2405 p^{4} T^{6} + 58 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 12 T + 8318 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 318 T + 51257 T^{2} - 5580582 T^{3} + 459199092 T^{4} - 5580582 p^{2} T^{5} + 51257 p^{4} T^{6} - 318 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 110 T - 2957 T^{2} - 283250 T^{3} + 112247068 T^{4} - 283250 p^{2} T^{5} - 2957 p^{4} T^{6} - 110 p^{6} T^{7} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 380 T^{2} + 89625894 T^{4} + 380 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 378 T + 71921 T^{2} + 9182754 T^{3} + 904668996 T^{4} + 9182754 p^{2} T^{5} + 71921 p^{4} T^{6} + 378 p^{6} T^{7} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 26620 T^{2} + 330657414 T^{4} - 26620 p^{4} T^{6} + p^{8} T^{8} \) |
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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
−15.22279019022005097097799665439, −14.49967147108960395923192122546, −14.02141981904875035769810097803, −13.89178078181923680281174341622, −13.75515344716880117872314244192, −12.60275105804830871495238879448, −12.57998488392336310216795537851, −12.16506698103550209906162500864, −11.78489684476757193141817960336, −11.45939919926564139447325209898, −11.09874051596798499778823202451, −10.99442918404639935465779445083, −10.68917498998943417116922782835, −9.908535361913521390313667049692, −9.012872723719945435784824270220, −8.967667850166112942040191886460, −8.625141808881862966633483659949, −7.75089215970672930435832883388, −7.38191162710491748042669853723, −6.59018977696893865850250341187, −6.41286003038764665169598384020, −5.53199225414551369734500404159, −5.02126322441029586578365387396, −4.48579382903562604163237868559, −3.81523649303225126491768400041,
3.81523649303225126491768400041, 4.48579382903562604163237868559, 5.02126322441029586578365387396, 5.53199225414551369734500404159, 6.41286003038764665169598384020, 6.59018977696893865850250341187, 7.38191162710491748042669853723, 7.75089215970672930435832883388, 8.625141808881862966633483659949, 8.967667850166112942040191886460, 9.012872723719945435784824270220, 9.908535361913521390313667049692, 10.68917498998943417116922782835, 10.99442918404639935465779445083, 11.09874051596798499778823202451, 11.45939919926564139447325209898, 11.78489684476757193141817960336, 12.16506698103550209906162500864, 12.57998488392336310216795537851, 12.60275105804830871495238879448, 13.75515344716880117872314244192, 13.89178078181923680281174341622, 14.02141981904875035769810097803, 14.49967147108960395923192122546, 15.22279019022005097097799665439
