| L(s) = 1 | − 6.25e3·5-s + 3.34e3·7-s − 2.98e5·11-s + 1.20e6·13-s − 9.05e6·17-s + 1.94e7·19-s − 5.59e7·23-s + 2.92e7·25-s − 4.18e7·29-s + 6.22e7·31-s − 2.08e7·35-s − 7.29e8·37-s + 6.08e8·41-s − 1.12e9·43-s + 1.65e9·47-s − 1.64e9·49-s − 5.72e9·53-s + 1.86e9·55-s − 3.75e9·59-s + 4.92e9·61-s − 7.56e9·65-s − 5.84e9·67-s + 4.33e10·71-s + 3.36e10·73-s − 9.96e8·77-s − 5.47e10·79-s + 9.30e9·83-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.0751·7-s − 0.558·11-s + 0.903·13-s − 1.54·17-s + 1.80·19-s − 1.81·23-s + 3/5·25-s − 0.378·29-s + 0.390·31-s − 0.0671·35-s − 1.72·37-s + 0.820·41-s − 1.17·43-s + 1.05·47-s − 0.831·49-s − 1.88·53-s + 0.499·55-s − 0.684·59-s + 0.746·61-s − 0.808·65-s − 0.528·67-s + 2.85·71-s + 1.89·73-s − 0.0419·77-s − 2.00·79-s + 0.259·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(2.150582281\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.150582281\) |
| \(L(\frac{13}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{5} T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 3340 T + 236350050 p T^{2} - 3340 p^{11} T^{3} + p^{22} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 27120 p T + 564442123222 T^{2} + 27120 p^{12} T^{3} + p^{22} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 1209820 T + 2088490732398 T^{2} - 1209820 p^{11} T^{3} + p^{22} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 9056340 T + 84322733142022 T^{2} + 9056340 p^{11} T^{3} + p^{22} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 19439368 T + 284381272377894 T^{2} - 19439368 p^{11} T^{3} + p^{22} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 55926420 T + 116503185471650 p T^{2} + 55926420 p^{11} T^{3} + p^{22} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 41841708 T + 3630990695891374 T^{2} + 41841708 p^{11} T^{3} + p^{22} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 62230792 T - 13999135099203522 T^{2} - 62230792 p^{11} T^{3} + p^{22} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 729235940 T + 388817795213444670 T^{2} + 729235940 p^{11} T^{3} + p^{22} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 608419068 T + 1119141666750688438 T^{2} - 608419068 p^{11} T^{3} + p^{22} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 1129440740 T + 1605906671305010214 T^{2} + 1129440740 p^{11} T^{3} + p^{22} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 1653072900 T + 3734132615108598142 T^{2} - 1653072900 p^{11} T^{3} + p^{22} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 5724887340 T + 23865188747531536510 T^{2} + 5724887340 p^{11} T^{3} + p^{22} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 3756433896 T + 50973876172267502422 T^{2} + 3756433896 p^{11} T^{3} + p^{22} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4923703564 T + 12712509351121558446 T^{2} - 4923703564 p^{11} T^{3} + p^{22} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 5843244140 T + \)\(24\!\cdots\!10\)\( T^{2} + 5843244140 p^{11} T^{3} + p^{22} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 43352162664 T + \)\(91\!\cdots\!66\)\( T^{2} - 43352162664 p^{11} T^{3} + p^{22} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 33647099620 T + \)\(88\!\cdots\!78\)\( T^{2} - 33647099620 p^{11} T^{3} + p^{22} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 54799425296 T + \)\(19\!\cdots\!62\)\( T^{2} + 54799425296 p^{11} T^{3} + p^{22} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 9303032100 T + \)\(25\!\cdots\!70\)\( T^{2} - 9303032100 p^{11} T^{3} + p^{22} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 3688968372 T + \)\(27\!\cdots\!74\)\( T^{2} + 3688968372 p^{11} T^{3} + p^{22} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 65892157780 T + \)\(13\!\cdots\!30\)\( T^{2} - 65892157780 p^{11} T^{3} + p^{22} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.77318930395714821153684173699, −10.68295538495868104760242813157, −9.698190947377033977862229143363, −9.565941509209725068021975967829, −8.696818918420324124784902453044, −8.415027683375466960694468544910, −7.76537146123162016448047898576, −7.59051468460137765033158239482, −6.65340090961756173288755158713, −6.50971203044327298237043946871, −5.59340770927657632448115906344, −5.21655280185413850946191061422, −4.46506218759051644178017186188, −4.07261614956875472569105784425, −3.29553811917853618227821746041, −3.12515727751054072485498249771, −1.93448339632033880133435966067, −1.83844017287697153594227038202, −0.69468499115170986567654686805, −0.42288162974959194966895113781,
0.42288162974959194966895113781, 0.69468499115170986567654686805, 1.83844017287697153594227038202, 1.93448339632033880133435966067, 3.12515727751054072485498249771, 3.29553811917853618227821746041, 4.07261614956875472569105784425, 4.46506218759051644178017186188, 5.21655280185413850946191061422, 5.59340770927657632448115906344, 6.50971203044327298237043946871, 6.65340090961756173288755158713, 7.59051468460137765033158239482, 7.76537146123162016448047898576, 8.415027683375466960694468544910, 8.696818918420324124784902453044, 9.565941509209725068021975967829, 9.698190947377033977862229143363, 10.68295538495868104760242813157, 10.77318930395714821153684173699
