| L(s) = 1 | + 15·5-s − 10·7-s + 28·11-s − 78·13-s + 11·17-s − 71·19-s − 25·23-s + 150·25-s − 118·29-s − 107·31-s − 150·35-s − 410·37-s − 592·41-s + 52·43-s − 580·47-s − 225·49-s − 169·53-s + 420·55-s + 234·59-s − 673·61-s − 1.17e3·65-s + 386·67-s + 16·71-s − 892·73-s − 280·77-s + 1.26e3·79-s − 1.81e3·83-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 0.539·7-s + 0.767·11-s − 1.66·13-s + 0.156·17-s − 0.857·19-s − 0.226·23-s + 6/5·25-s − 0.755·29-s − 0.619·31-s − 0.724·35-s − 1.82·37-s − 2.25·41-s + 0.184·43-s − 1.80·47-s − 0.655·49-s − 0.437·53-s + 1.02·55-s + 0.516·59-s − 1.41·61-s − 2.23·65-s + 0.703·67-s + 0.0267·71-s − 1.43·73-s − 0.414·77-s + 1.79·79-s − 2.40·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 + 10 T + 325 T^{2} + 4052 T^{3} + 325 p^{3} T^{4} + 10 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 28 T + 111 p T^{2} - 66112 T^{3} + 111 p^{4} T^{4} - 28 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 6 p T + 579 p T^{2} + 336036 T^{3} + 579 p^{4} T^{4} + 6 p^{7} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 11 T + 9706 T^{2} + 12229 T^{3} + 9706 p^{3} T^{4} - 11 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 71 T + 19688 T^{2} + 915683 T^{3} + 19688 p^{3} T^{4} + 71 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 25 T + 2176 T^{2} - 755303 T^{3} + 2176 p^{3} T^{4} + 25 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 118 T + 15375 T^{2} + 3162436 T^{3} + 15375 p^{3} T^{4} + 118 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 107 T + 26244 T^{2} + 9258271 T^{3} + 26244 p^{3} T^{4} + 107 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 410 T + 126819 T^{2} + 26481628 T^{3} + 126819 p^{3} T^{4} + 410 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 592 T + 301279 T^{2} + 2091704 p T^{3} + 301279 p^{3} T^{4} + 592 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 52 T + 175069 T^{2} - 917504 T^{3} + 175069 p^{3} T^{4} - 52 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 580 T + 309773 T^{2} + 92211384 T^{3} + 309773 p^{3} T^{4} + 580 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 169 T + 419518 T^{2} + 50902417 T^{3} + 419518 p^{3} T^{4} + 169 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 234 T + 385665 T^{2} - 30035844 T^{3} + 385665 p^{3} T^{4} - 234 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 673 T + 749898 T^{2} + 301087949 T^{3} + 749898 p^{3} T^{4} + 673 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 386 T + 549197 T^{2} - 181189028 T^{3} + 549197 p^{3} T^{4} - 386 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 16 T + 828849 T^{2} - 56595352 T^{3} + 828849 p^{3} T^{4} - 16 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 892 T + 466231 T^{2} + 343918592 T^{3} + 466231 p^{3} T^{4} + 892 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 1263 T + 1118004 T^{2} - 741292675 T^{3} + 1118004 p^{3} T^{4} - 1263 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 1815 T + 2216940 T^{2} + 2103744111 T^{3} + 2216940 p^{3} T^{4} + 1815 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 1800 T + 2584479 T^{2} + 2568723640 T^{3} + 2584479 p^{3} T^{4} + 1800 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 840 T + 1820643 T^{2} + 757404304 T^{3} + 1820643 p^{3} T^{4} + 840 p^{6} T^{5} + p^{9} T^{6} \) |
| 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
−9.035664958008461470423023596572, −8.461115227342521391028209893421, −8.270344110527583338924910666038, −8.248523977254044795457031204903, −7.52345318262806028737014518240, −7.41045418160539047043004649660, −7.06398477621539594801221273145, −6.62656477442392427031115104198, −6.60113888884886237196029596081, −6.52251549683500906326761494322, −5.87865217134129348481622521468, −5.60735474978539539723798262400, −5.47002637643478913287444697716, −5.01301152546786692070357329725, −4.71962788640532675026696984438, −4.67647807200302959928775533320, −3.92601991362238562594339022162, −3.59741262856072851520863259276, −3.55871896876095749881010739858, −2.78189170324357191535722802232, −2.69345032265620194399008190670, −2.30172847567971195882660939865, −1.69285731331916044775571755629, −1.45897983220054615474409706402, −1.38965178304697007375410027181, 0, 0, 0,
1.38965178304697007375410027181, 1.45897983220054615474409706402, 1.69285731331916044775571755629, 2.30172847567971195882660939865, 2.69345032265620194399008190670, 2.78189170324357191535722802232, 3.55871896876095749881010739858, 3.59741262856072851520863259276, 3.92601991362238562594339022162, 4.67647807200302959928775533320, 4.71962788640532675026696984438, 5.01301152546786692070357329725, 5.47002637643478913287444697716, 5.60735474978539539723798262400, 5.87865217134129348481622521468, 6.52251549683500906326761494322, 6.60113888884886237196029596081, 6.62656477442392427031115104198, 7.06398477621539594801221273145, 7.41045418160539047043004649660, 7.52345318262806028737014518240, 8.248523977254044795457031204903, 8.270344110527583338924910666038, 8.461115227342521391028209893421, 9.035664958008461470423023596572
