| L(s) = 1 | − 6.25e3·5-s − 4.45e4·7-s + 2.44e5·11-s − 1.68e6·13-s + 5.91e6·17-s − 1.91e7·19-s − 4.83e6·23-s + 2.92e7·25-s − 6.16e7·29-s − 3.88e8·31-s + 2.78e8·35-s + 1.71e8·37-s − 4.94e8·41-s − 8.76e8·43-s − 9.26e8·47-s − 1.88e9·49-s + 7.51e9·53-s − 1.52e9·55-s + 1.77e10·59-s − 1.18e10·61-s + 1.05e10·65-s − 1.49e10·67-s + 5.86e9·71-s − 3.19e10·73-s − 1.08e10·77-s + 6.66e9·79-s + 1.34e10·83-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.00·7-s + 0.457·11-s − 1.25·13-s + 1.01·17-s − 1.77·19-s − 0.156·23-s + 3/5·25-s − 0.557·29-s − 2.43·31-s + 0.895·35-s + 0.405·37-s − 0.665·41-s − 0.909·43-s − 0.589·47-s − 0.953·49-s + 2.46·53-s − 0.409·55-s + 3.22·59-s − 1.79·61-s + 1.12·65-s − 1.35·67-s + 0.385·71-s − 1.80·73-s − 0.458·77-s + 0.243·79-s + 0.374·83-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
| \(L(6)\) |
\(\approx\) |
\(0.008291838743\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.008291838743\) |
| \(L(\frac{13}{2})\) |
|
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 | $D_{4}$ | \( 1 + 44504 T + 78887310 p^{2} T^{2} + 44504 p^{11} T^{3} + p^{22} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 244488 T - 157282652042 T^{2} - 244488 p^{11} T^{3} + p^{22} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 129500 p T + 2971603683774 T^{2} + 129500 p^{12} T^{3} + p^{22} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 5918724 T + 77005347089110 T^{2} - 5918724 p^{11} T^{3} + p^{22} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 19147520 T + 262891683344838 T^{2} + 19147520 p^{11} T^{3} + p^{22} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4834104 T - 842086473014642 T^{2} + 4834104 p^{11} T^{3} + p^{22} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 61623852 T + 1586286373961134 T^{2} + 61623852 p^{11} T^{3} + p^{22} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 388179752 T + 88293666278267838 T^{2} + 388179752 p^{11} T^{3} + p^{22} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 171134788 T + 300059711927738862 T^{2} - 171134788 p^{11} T^{3} + p^{22} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 494016420 T + 1021540088843048182 T^{2} + 494016420 p^{11} T^{3} + p^{22} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 876600536 T + 776312017876968438 T^{2} + 876600536 p^{11} T^{3} + p^{22} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 926495544 T + 4609971128143087390 T^{2} + 926495544 p^{11} T^{3} + p^{22} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 7512649332 T + 31907881739466274750 T^{2} - 7512649332 p^{11} T^{3} + p^{22} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 17730138888 T + \)\(13\!\cdots\!54\)\( T^{2} - 17730138888 p^{11} T^{3} + p^{22} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 11850180356 T + 94666487881205008206 T^{2} + 11850180356 p^{11} T^{3} + p^{22} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 14955920936 T + \)\(21\!\cdots\!90\)\( T^{2} + 14955920936 p^{11} T^{3} + p^{22} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 5864548560 T + \)\(43\!\cdots\!42\)\( T^{2} - 5864548560 p^{11} T^{3} + p^{22} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 31931517308 T + \)\(87\!\cdots\!70\)\( T^{2} + 31931517308 p^{11} T^{3} + p^{22} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6664632904 T - \)\(41\!\cdots\!38\)\( T^{2} - 6664632904 p^{11} T^{3} + p^{22} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 13423662696 T + \)\(98\!\cdots\!38\)\( T^{2} - 13423662696 p^{11} T^{3} + p^{22} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 3984712860 T + \)\(51\!\cdots\!78\)\( T^{2} - 3984712860 p^{11} T^{3} + p^{22} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 7633999012 T + \)\(64\!\cdots\!42\)\( T^{2} - 7633999012 p^{11} T^{3} + p^{22} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.84602929968469146680726085481, −10.25186307314836654648126204955, −10.01391120836216730149907710703, −9.393836044548736521539813892559, −8.837586159808085524971596667083, −8.504080241240133608032976787898, −7.63408530508134689842855888762, −7.48716661465661917000736050339, −6.66074622745906932790982071231, −6.60381567230066542947281475873, −5.45600270886978499879380228977, −5.43412791738999378776075499011, −4.28966403278931981733621132675, −4.12672727068591881019049331577, −3.35691456419600780129950498491, −3.05059447342319850918397439185, −2.11880385072129264923030505678, −1.73728998249271011277746205228, −0.73572541031471314422596488225, −0.02430287989286163849785427180,
0.02430287989286163849785427180, 0.73572541031471314422596488225, 1.73728998249271011277746205228, 2.11880385072129264923030505678, 3.05059447342319850918397439185, 3.35691456419600780129950498491, 4.12672727068591881019049331577, 4.28966403278931981733621132675, 5.43412791738999378776075499011, 5.45600270886978499879380228977, 6.60381567230066542947281475873, 6.66074622745906932790982071231, 7.48716661465661917000736050339, 7.63408530508134689842855888762, 8.504080241240133608032976787898, 8.837586159808085524971596667083, 9.393836044548736521539813892559, 10.01391120836216730149907710703, 10.25186307314836654648126204955, 10.84602929968469146680726085481
