| L(s) = 1 | − 9·3-s − 10·5-s + 14·7-s + 54·9-s − 52·13-s + 90·15-s + 26·17-s − 28·19-s − 126·21-s + 164·23-s − 111·25-s − 270·27-s − 174·29-s + 318·31-s − 140·35-s − 296·37-s + 468·39-s − 118·41-s + 260·43-s − 540·45-s + 204·47-s − 253·49-s − 234·51-s − 1.08e3·53-s + 252·57-s − 196·59-s − 1.53e3·61-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.894·5-s + 0.755·7-s + 2·9-s − 1.10·13-s + 1.54·15-s + 0.370·17-s − 0.338·19-s − 1.30·21-s + 1.48·23-s − 0.887·25-s − 1.92·27-s − 1.11·29-s + 1.84·31-s − 0.676·35-s − 1.31·37-s + 1.92·39-s − 0.449·41-s + 0.922·43-s − 1.78·45-s + 0.633·47-s − 0.737·49-s − 0.642·51-s − 2.81·53-s + 0.585·57-s − 0.432·59-s − 3.22·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 + 2 p T + 211 T^{2} + 2396 T^{3} + 211 p^{3} T^{4} + 2 p^{7} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 2 p T + 449 T^{2} - 3788 T^{3} + 449 p^{3} T^{4} - 2 p^{7} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 107 p T^{2} - 49152 T^{3} + 107 p^{4} T^{4} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 4 p T + 5487 T^{2} + 172616 T^{3} + 5487 p^{3} T^{4} + 4 p^{7} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 26 T + 3615 T^{2} + 222100 T^{3} + 3615 p^{3} T^{4} - 26 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 28 T + 9489 T^{2} + 658856 T^{3} + 9489 p^{3} T^{4} + 28 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 164 T + 42885 T^{2} - 4036280 T^{3} + 42885 p^{3} T^{4} - 164 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 6 p T + 77131 T^{2} + 8426004 T^{3} + 77131 p^{3} T^{4} + 6 p^{7} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 318 T + 93849 T^{2} - 15197452 T^{3} + 93849 p^{3} T^{4} - 318 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 8 p T + 105879 T^{2} + 29903632 T^{3} + 105879 p^{3} T^{4} + 8 p^{7} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 118 T + 89463 T^{2} - 3720620 T^{3} + 89463 p^{3} T^{4} + 118 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 260 T + 157545 T^{2} - 32742232 T^{3} + 157545 p^{3} T^{4} - 260 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 204 T + 283677 T^{2} - 40395048 T^{3} + 283677 p^{3} T^{4} - 204 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 1086 T + 736099 T^{2} + 344026068 T^{3} + 736099 p^{3} T^{4} + 1086 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 196 T + 566137 T^{2} + 71984984 T^{3} + 566137 p^{3} T^{4} + 196 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 1536 T + 1409583 T^{2} + 802126848 T^{3} + 1409583 p^{3} T^{4} + 1536 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 660 T + 592833 T^{2} - 364778232 T^{3} + 592833 p^{3} T^{4} - 660 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 12 p T + 1006773 T^{2} + 524795352 T^{3} + 1006773 p^{3} T^{4} + 12 p^{7} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 478 T + 911095 T^{2} + 251066948 T^{3} + 911095 p^{3} T^{4} + 478 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 22 T + 1407593 T^{2} - 13791100 T^{3} + 1407593 p^{3} T^{4} - 22 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 1136 T + 2100385 T^{2} + 1337053600 T^{3} + 2100385 p^{3} T^{4} + 1136 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 110 T + 2073543 T^{2} - 156516836 T^{3} + 2073543 p^{3} T^{4} - 110 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 1222 T + 2989679 T^{2} + 2155770388 T^{3} + 2989679 p^{3} T^{4} + 1222 p^{6} T^{5} + p^{9} T^{6} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.545926400828767969814625644481, −8.890437072375400762597956535962, −8.590788082878218212723863092108, −8.361679690073748069119438934250, −7.84658083261436586516033190534, −7.70353582395919625639268348945, −7.52801820765670183269783990435, −7.21522431960868665290002735993, −6.75365091022357319278892968513, −6.73429815944415528067111399346, −6.12082521939549501365940537515, −5.93016017244913180203494266778, −5.75003461029218146942210396831, −5.02673117631291495153529473645, −5.00723955911273394122720965364, −4.86426477951626204554159407993, −4.43426542191552671661267341898, −3.96262756695492195544689232680, −3.92624868776818474894538527610, −3.08474330877103364898518681223, −2.96200964001524228068065483598, −2.41312144897859229375571832259, −1.69030753949563596705831871340, −1.31890091355433483197921948879, −1.18864037236491074372327602677, 0, 0, 0,
1.18864037236491074372327602677, 1.31890091355433483197921948879, 1.69030753949563596705831871340, 2.41312144897859229375571832259, 2.96200964001524228068065483598, 3.08474330877103364898518681223, 3.92624868776818474894538527610, 3.96262756695492195544689232680, 4.43426542191552671661267341898, 4.86426477951626204554159407993, 5.00723955911273394122720965364, 5.02673117631291495153529473645, 5.75003461029218146942210396831, 5.93016017244913180203494266778, 6.12082521939549501365940537515, 6.73429815944415528067111399346, 6.75365091022357319278892968513, 7.21522431960868665290002735993, 7.52801820765670183269783990435, 7.70353582395919625639268348945, 7.84658083261436586516033190534, 8.361679690073748069119438934250, 8.590788082878218212723863092108, 8.890437072375400762597956535962, 9.545926400828767969814625644481
