| L(s) = 1 | + 6·3-s + 8·5-s − 16·7-s + 27·9-s + 8·11-s − 72·13-s + 48·15-s − 36·17-s − 136·19-s − 96·21-s − 256·23-s + 6·25-s + 108·27-s − 152·29-s + 80·31-s + 48·33-s − 128·35-s − 136·37-s − 432·39-s − 436·41-s + 712·43-s + 216·45-s + 224·47-s − 286·49-s − 216·51-s − 344·53-s + 64·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.715·5-s − 0.863·7-s + 9-s + 0.219·11-s − 1.53·13-s + 0.826·15-s − 0.513·17-s − 1.64·19-s − 0.997·21-s − 2.32·23-s + 0.0479·25-s + 0.769·27-s − 0.973·29-s + 0.463·31-s + 0.253·33-s − 0.618·35-s − 0.604·37-s − 1.77·39-s − 1.66·41-s + 2.52·43-s + 0.715·45-s + 0.695·47-s − 0.833·49-s − 0.593·51-s − 0.891·53-s + 0.156·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 8 T + 58 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 16 T + 542 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 8 T - 650 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 72 T + 4858 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 36 T + 6822 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 136 T + 15014 T^{2} + 136 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 256 T + 39886 T^{2} + 256 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 152 T + 37706 T^{2} + 152 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 80 T + 26030 T^{2} - 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 136 T + 102602 T^{2} + 136 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 436 T + 102166 T^{2} + 436 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 712 T + 282422 T^{2} - 712 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 224 T + 179422 T^{2} - 224 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 344 T + 280538 T^{2} + 344 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 324 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 324 T + p^{3} T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 456 T + 174278 T^{2} + 456 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 2048 T + 1763566 T^{2} + 2048 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 660 T + 407702 T^{2} - 660 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 496 T + 323534 T^{2} + 496 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 776 T + 1131046 T^{2} + 776 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 532 T + 1467382 T^{2} - 532 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1220 T + 586694 T^{2} + 1220 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.549810007803317865183701168653, −9.417269832898761271705241585514, −8.884187353407915566513531552430, −8.604208058845822426859242814313, −7.82607428831707766330780378813, −7.81253267477423767725633207067, −7.04826880203249416803264817137, −6.77239688162782527179376262181, −6.14049214730816856472003521919, −5.92102413191816176825145186277, −5.24561736223728480055584583523, −4.43776455874893329595709365575, −4.21176991604164993014959170122, −3.72248849025168952421179160265, −2.84290769584984853851837647939, −2.59312506165921094393055108724, −1.93707007123723651046221997038, −1.61998869897830653345841104550, 0, 0,
1.61998869897830653345841104550, 1.93707007123723651046221997038, 2.59312506165921094393055108724, 2.84290769584984853851837647939, 3.72248849025168952421179160265, 4.21176991604164993014959170122, 4.43776455874893329595709365575, 5.24561736223728480055584583523, 5.92102413191816176825145186277, 6.14049214730816856472003521919, 6.77239688162782527179376262181, 7.04826880203249416803264817137, 7.81253267477423767725633207067, 7.82607428831707766330780378813, 8.604208058845822426859242814313, 8.884187353407915566513531552430, 9.417269832898761271705241585514, 9.549810007803317865183701168653
