| L(s) = 1 | + 4.48e3·9-s + 2.05e5·11-s − 2.74e5·19-s + 1.37e7·29-s + 5.83e5·31-s + 5.95e7·41-s − 3.12e7·49-s + 1.85e8·59-s + 3.91e8·61-s + 6.22e8·71-s − 1.08e9·79-s + 6.07e8·81-s + 9.24e8·89-s + 9.21e8·99-s − 2.37e9·101-s + 2.90e9·109-s + 1.69e10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.35e10·169-s + ⋯ |
| L(s) = 1 | + 0.227·9-s + 4.23·11-s − 0.483·19-s + 3.61·29-s + 0.113·31-s + 3.29·41-s − 0.773·49-s + 1.99·59-s + 3.61·61-s + 2.90·71-s − 3.13·79-s + 1.56·81-s + 1.56·89-s + 0.963·99-s − 2.26·101-s + 1.97·109-s + 7.20·121-s + 1.27·169-s − 0.110·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(21.04604251\) |
| \(L(\frac12)\) |
\(\approx\) |
\(21.04604251\) |
| \(L(\frac{11}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 4484 T^{2} - 65252762 p^{2} T^{4} - 4484 p^{18} T^{6} + p^{36} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 31230700 T^{2} + 71158420123302 p^{2} T^{4} + 31230700 p^{18} T^{6} + p^{36} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 102720 T + 7335543382 T^{2} - 102720 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 13506080684 T^{2} + 64066498776709104822 T^{4} - 13506080684 p^{18} T^{6} + p^{36} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 419518771196 T^{2} + \)\(71\!\cdots\!22\)\( T^{4} - 419518771196 p^{18} T^{6} + p^{36} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 137272 T + 111610161654 T^{2} + 137272 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 5330064218900 T^{2} + \)\(13\!\cdots\!38\)\( T^{4} - 5330064218900 p^{18} T^{6} + p^{36} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 6893748 T + 40195999658014 T^{2} - 6893748 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 291832 T + 38964935800398 T^{2} - 291832 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 310941286370060 T^{2} + \)\(48\!\cdots\!58\)\( T^{4} - 310941286370060 p^{18} T^{6} + p^{36} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 29773452 T + 771012402449398 T^{2} - 29773452 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 1540458208886372 T^{2} + \)\(10\!\cdots\!94\)\( T^{4} - 1540458208886372 p^{18} T^{6} + p^{36} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 2450505224578676 T^{2} + \)\(38\!\cdots\!22\)\( T^{4} - 2450505224578676 p^{18} T^{6} + p^{36} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 11327676942925580 T^{2} + \)\(53\!\cdots\!78\)\( T^{4} - 11327676942925580 p^{18} T^{6} + p^{36} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 92930856 T + 16656477955483462 T^{2} - 92930856 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 195673924 T + 22434263296171326 T^{2} - 195673924 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 83008173082523780 T^{2} + \)\(31\!\cdots\!18\)\( T^{4} - 83008173082523780 p^{18} T^{6} + p^{36} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 311207016 T + 76405636625293726 T^{2} - 311207016 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 60959480039527964 T^{2} + \)\(70\!\cdots\!62\)\( T^{4} - 60959480039527964 p^{18} T^{6} + p^{36} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 542261776 T + 313115996157615582 T^{2} + 542261776 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 43663493653967740 T^{2} + \)\(69\!\cdots\!18\)\( T^{4} + 43663493653967740 p^{18} T^{6} + p^{36} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 462291852 T + 159603168035249494 T^{2} - 462291852 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 1302744298917711740 T^{2} + \)\(11\!\cdots\!78\)\( T^{4} - 1302744298917711740 p^{18} T^{6} + p^{36} T^{8} \) |
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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
−8.469583959061583603084619033485, −8.098657821122497039631896594684, −7.977651016796078644147375176366, −7.34412420514604666718390365178, −7.04274304739627712749868028358, −6.68140927959006282084802932387, −6.48959362326266919930338897695, −6.47227501948201427336652770684, −6.30493506223379429858111601372, −5.52110894303943646844059880741, −5.45871591424117078036100481359, −4.92956534715695437065149202661, −4.39490308523010856299416834566, −4.07777571561675816803500289952, −4.02361708428339548886355441616, −4.01940421334872634020396089720, −3.22924862882152045191048659691, −2.97436888865915693018781726461, −2.52054346068563088058166648823, −1.96053991166104803637153222117, −1.85963845756409722683320909958, −1.19500234071844561121516419736, −0.846345034071271014751409264547, −0.790435777316921228349370026939, −0.66808915609211207551638267930,
0.66808915609211207551638267930, 0.790435777316921228349370026939, 0.846345034071271014751409264547, 1.19500234071844561121516419736, 1.85963845756409722683320909958, 1.96053991166104803637153222117, 2.52054346068563088058166648823, 2.97436888865915693018781726461, 3.22924862882152045191048659691, 4.01940421334872634020396089720, 4.02361708428339548886355441616, 4.07777571561675816803500289952, 4.39490308523010856299416834566, 4.92956534715695437065149202661, 5.45871591424117078036100481359, 5.52110894303943646844059880741, 6.30493506223379429858111601372, 6.47227501948201427336652770684, 6.48959362326266919930338897695, 6.68140927959006282084802932387, 7.04274304739627712749868028358, 7.34412420514604666718390365178, 7.977651016796078644147375176366, 8.098657821122497039631896594684, 8.469583959061583603084619033485
