| L(s) = 1 | − 2-s − 9·4-s + 15·5-s − 25·7-s + 21·8-s − 15·10-s − 58·11-s − 47·13-s + 25·14-s − 13·16-s − 34·17-s − 5·19-s − 135·20-s + 58·22-s + 51·23-s + 150·25-s + 47·26-s + 225·28-s − 350·29-s + 638·31-s − 199·32-s + 34·34-s − 375·35-s − 414·37-s + 5·38-s + 315·40-s − 179·41-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 9/8·4-s + 1.34·5-s − 1.34·7-s + 0.928·8-s − 0.474·10-s − 1.58·11-s − 1.00·13-s + 0.477·14-s − 0.203·16-s − 0.485·17-s − 0.0603·19-s − 1.50·20-s + 0.562·22-s + 0.462·23-s + 6/5·25-s + 0.354·26-s + 1.51·28-s − 2.24·29-s + 3.69·31-s − 1.09·32-s + 0.171·34-s − 1.81·35-s − 1.83·37-s + 0.0213·38-s + 1.24·40-s − 0.681·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66430125 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66430125 ^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + T + 5 p T^{2} - p T^{3} + 5 p^{4} T^{4} + p^{6} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 25 T + 381 T^{2} + 1010 T^{3} + 381 p^{3} T^{4} + 25 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 58 T + 3082 T^{2} + 157396 T^{3} + 3082 p^{3} T^{4} + 58 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 47 T - 841 T^{2} - 163834 T^{3} - 841 p^{3} T^{4} + 47 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 2 p T + 12031 T^{2} + 14300 p T^{3} + 12031 p^{3} T^{4} + 2 p^{7} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 5 T + 9800 T^{2} - 231055 T^{3} + 9800 p^{3} T^{4} + 5 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 51 T + 21261 T^{2} - 199878 T^{3} + 21261 p^{3} T^{4} - 51 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 350 T + 53104 T^{2} + 5834540 T^{3} + 53104 p^{3} T^{4} + 350 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 638 T + 224694 T^{2} - 47552380 T^{3} + 224694 p^{3} T^{4} - 638 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 414 T + 167991 T^{2} + 41362924 T^{3} + 167991 p^{3} T^{4} + 414 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 179 T + 54946 T^{2} + 6874091 T^{3} + 54946 p^{3} T^{4} + 179 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 836 T + 459353 T^{2} + 151628696 T^{3} + 459353 p^{3} T^{4} + 836 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 5 p T + 241789 T^{2} + 39791434 T^{3} + 241789 p^{3} T^{4} + 5 p^{7} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 505 T + 481759 T^{2} + 148865086 T^{3} + 481759 p^{3} T^{4} + 505 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 535 T + 364888 T^{2} + 242073187 T^{3} + 364888 p^{3} T^{4} + 535 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 104 T + 386531 T^{2} + 23669216 T^{3} + 386531 p^{3} T^{4} + 104 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 40 T + 527109 T^{2} + 8344832 T^{3} + 527109 p^{3} T^{4} + 40 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 452 T + 808810 T^{2} + 207368090 T^{3} + 808810 p^{3} T^{4} + 452 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 710 T + 1311287 T^{2} + 561111668 T^{3} + 1311287 p^{3} T^{4} + 710 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 634 T + 1358829 T^{2} + 630226508 T^{3} + 1358829 p^{3} T^{4} + 634 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 1734 T + 1997313 T^{2} + 1760622468 T^{3} + 1997313 p^{3} T^{4} + 1734 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 852 T + 2335632 T^{2} - 1219193610 T^{3} + 2335632 p^{3} T^{4} - 852 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 1162851 T^{2} - 703275008 T^{3} + 1162851 p^{3} T^{4} + p^{9} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.16478140631094759944993060851, −9.636686395186789825868077336809, −9.551803346719183138579553726493, −9.459405219447065759895262686290, −8.726889212965877575502493346170, −8.686089915407460075227237399907, −8.444924049108022118121680814566, −7.82325278058687865780446799668, −7.79613708319798334569597549043, −7.14811395945593464516058885345, −6.87759025341050580412273402887, −6.58441733249264846757167061942, −6.37039664675292635470370550324, −5.83845888550232078617606521778, −5.33496753181973848550727468372, −5.25270629881691821845581184062, −4.86228557131038574859440496527, −4.54237634283927126290037434941, −4.22897346335225905324725636687, −3.34437935696238541964785295487, −3.24236934524999340222888111339, −2.64268932029356649570775336587, −2.44331582793314423568838919208, −1.54718867321210697574949638969, −1.47157720848357479395799375154, 0, 0, 0,
1.47157720848357479395799375154, 1.54718867321210697574949638969, 2.44331582793314423568838919208, 2.64268932029356649570775336587, 3.24236934524999340222888111339, 3.34437935696238541964785295487, 4.22897346335225905324725636687, 4.54237634283927126290037434941, 4.86228557131038574859440496527, 5.25270629881691821845581184062, 5.33496753181973848550727468372, 5.83845888550232078617606521778, 6.37039664675292635470370550324, 6.58441733249264846757167061942, 6.87759025341050580412273402887, 7.14811395945593464516058885345, 7.79613708319798334569597549043, 7.82325278058687865780446799668, 8.444924049108022118121680814566, 8.686089915407460075227237399907, 8.726889212965877575502493346170, 9.459405219447065759895262686290, 9.551803346719183138579553726493, 9.636686395186789825868077336809, 10.16478140631094759944993060851
