| L(s) = 1 | + 18·3-s − 47·5-s + 174·7-s + 81·9-s − 407·11-s + 898·13-s − 846·15-s + 1.86e3·17-s − 1.46e3·19-s + 3.13e3·21-s − 44·23-s + 5.03e3·25-s − 1.45e3·27-s + 1.53e3·29-s − 1.11e4·31-s − 7.32e3·33-s − 8.17e3·35-s + 3.11e3·37-s + 1.61e4·39-s − 1.56e4·41-s + 2.52e4·43-s − 3.80e3·45-s + 9.57e3·47-s + 1.74e4·49-s + 3.36e4·51-s − 1.33e4·53-s + 1.91e4·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.840·5-s + 1.34·7-s + 1/3·9-s − 1.01·11-s + 1.47·13-s − 0.970·15-s + 1.56·17-s − 0.929·19-s + 1.54·21-s − 0.0173·23-s + 1.61·25-s − 0.384·27-s + 0.338·29-s − 2.08·31-s − 1.17·33-s − 1.12·35-s + 0.373·37-s + 1.70·39-s − 1.45·41-s + 2.08·43-s − 0.280·45-s + 0.632·47-s + 1.03·49-s + 1.81·51-s − 0.655·53-s + 0.852·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(6.006007998\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.006007998\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 174 T + 1837 p T^{2} - 174 p^{5} T^{3} + p^{10} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 47 T - 2823 T^{2} - 57246 T^{3} + 8652274 T^{4} - 57246 p^{5} T^{5} - 2823 p^{10} T^{6} + 47 p^{15} T^{7} + p^{20} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 37 p T - 196095 T^{2} + 1466754 p T^{3} + 78243359836 T^{4} + 1466754 p^{6} T^{5} - 196095 p^{10} T^{6} + 37 p^{16} T^{7} + p^{20} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 449 T + 748730 T^{2} - 449 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 1868 T - 194322 T^{2} - 1576651776 T^{3} + 6599507599699 T^{4} - 1576651776 p^{5} T^{5} - 194322 p^{10} T^{6} - 1868 p^{15} T^{7} + p^{20} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 77 p T + 1257499 T^{2} - 313338256 p T^{3} - 10651372361432 T^{4} - 313338256 p^{6} T^{5} + 1257499 p^{10} T^{6} + 77 p^{16} T^{7} + p^{20} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 44 T + 4831266 T^{2} - 778888704 T^{3} - 18116543724797 T^{4} - 778888704 p^{5} T^{5} + 4831266 p^{10} T^{6} + 44 p^{15} T^{7} + p^{20} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 767 T + 20137030 T^{2} - 767 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 11170 T + 46543337 T^{2} + 234204305370 T^{3} + 1727582741482268 T^{4} + 234204305370 p^{5} T^{5} + 46543337 p^{10} T^{6} + 11170 p^{15} T^{7} + p^{20} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 3113 T - 130639157 T^{2} - 5111583356 T^{3} + 14211907489914694 T^{4} - 5111583356 p^{5} T^{5} - 130639157 p^{10} T^{6} - 3113 p^{15} T^{7} + p^{20} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 7842 T + 247022914 T^{2} + 7842 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 12629 T + 333109116 T^{2} - 12629 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 9576 T - 8247254 p T^{2} - 197559583200 T^{3} + 156118328865021315 T^{4} - 197559583200 p^{5} T^{5} - 8247254 p^{11} T^{6} - 9576 p^{15} T^{7} + p^{20} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 13395 T - 607359157 T^{2} - 664469744580 T^{3} + 369108297267354750 T^{4} - 664469744580 p^{5} T^{5} - 607359157 p^{10} T^{6} + 13395 p^{15} T^{7} + p^{20} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 47521 T + 1032621423 T^{2} - 9704956266180 T^{3} - 623437058679047936 T^{4} - 9704956266180 p^{5} T^{5} + 1032621423 p^{10} T^{6} + 47521 p^{15} T^{7} + p^{20} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 63652 T + 1692210626 T^{2} - 42657907555152 T^{3} + 1431320230592549579 T^{4} - 42657907555152 p^{5} T^{5} + 1692210626 p^{10} T^{6} - 63652 p^{15} T^{7} + p^{20} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 44541 T - 1137929947 T^{2} - 18777613219974 T^{3} + 5244474546154615068 T^{4} - 18777613219974 p^{5} T^{5} - 1137929947 p^{10} T^{6} - 44541 p^{15} T^{7} + p^{20} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 125840 T + 7124822602 T^{2} - 125840 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 6039 T - 3120368275 T^{2} - 5974415250210 T^{3} + 5592395112555163014 T^{4} - 5974415250210 p^{5} T^{5} - 3120368275 p^{10} T^{6} + 6039 p^{15} T^{7} + p^{20} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 17588 T - 1195619881 T^{2} + 81769341182724 T^{3} - 8155228314208630432 T^{4} + 81769341182724 p^{5} T^{5} - 1195619881 p^{10} T^{6} - 17588 p^{15} T^{7} + p^{20} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 39325 T + 1663856032 T^{2} + 39325 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 83082 T - 5990581294 T^{2} + 143323189611840 T^{3} + 96508253465592493647 T^{4} + 143323189611840 p^{5} T^{5} - 5990581294 p^{10} T^{6} + 83082 p^{15} T^{7} + p^{20} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 1905 p T + 241552312 p T^{2} - 1905 p^{6} T^{3} + p^{10} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.76554414853755170578516733333, −7.28171789407181508001249805906, −7.20040467030098744738437459868, −6.93684886629802601050252035216, −6.49971569999435894891546937498, −6.27022547123651092358562959191, −5.84141885229580068617273337220, −5.66524850394053160899497326323, −5.47802455831414160356874381300, −4.90748163995433738743848955513, −4.88642637732233273015382290831, −4.75664410461319465885848909796, −4.13742844870521718234176784084, −3.87591368335653290594814474029, −3.56053634405247886481557408487, −3.43507755301673317499527790273, −3.30871963207071175035395504726, −2.72119513200681948576377328553, −2.19453782868535705895566271051, −2.18371727291491407590973261881, −1.98325521235115412132231605525, −1.11680117801672772281474728764, −1.10015129013693860208927195640, −0.796969532938272428600457014665, −0.24671663315313065664394785876,
0.24671663315313065664394785876, 0.796969532938272428600457014665, 1.10015129013693860208927195640, 1.11680117801672772281474728764, 1.98325521235115412132231605525, 2.18371727291491407590973261881, 2.19453782868535705895566271051, 2.72119513200681948576377328553, 3.30871963207071175035395504726, 3.43507755301673317499527790273, 3.56053634405247886481557408487, 3.87591368335653290594814474029, 4.13742844870521718234176784084, 4.75664410461319465885848909796, 4.88642637732233273015382290831, 4.90748163995433738743848955513, 5.47802455831414160356874381300, 5.66524850394053160899497326323, 5.84141885229580068617273337220, 6.27022547123651092358562959191, 6.49971569999435894891546937498, 6.93684886629802601050252035216, 7.20040467030098744738437459868, 7.28171789407181508001249805906, 7.76554414853755170578516733333
