| L(s) = 1 | + 156·5-s + 3.35e3·9-s − 2.64e3·11-s − 3.63e4·19-s + 9.31e4·25-s + 4.81e5·29-s − 1.61e5·31-s + 1.12e6·41-s + 5.23e5·45-s − 5.09e5·49-s − 4.11e5·55-s − 7.74e6·59-s − 4.30e6·61-s + 1.20e6·71-s − 5.71e6·79-s + 4.66e6·81-s + 1.36e7·89-s − 5.66e6·95-s − 8.85e6·99-s + 3.20e6·101-s − 5.07e7·109-s − 2.73e7·121-s + 3.74e7·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 0.558·5-s + 1.53·9-s − 0.598·11-s − 1.21·19-s + 1.19·25-s + 3.66·29-s − 0.972·31-s + 2.55·41-s + 0.856·45-s − 0.618·49-s − 0.333·55-s − 4.91·59-s − 2.42·61-s + 0.399·71-s − 1.30·79-s + 0.974·81-s + 2.05·89-s − 0.678·95-s − 0.917·99-s + 0.309·101-s − 3.75·109-s − 1.40·121-s + 1.71·125-s + 2.04·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(5.163802864\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.163802864\) |
| \(L(\frac{9}{2})\) |
|
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 | $D_{4}$ | \( 1 - 156 T - 2754 p^{2} T^{2} - 156 p^{7} T^{3} + p^{14} T^{4} \) |
| good | 3 | $C_2^2 \wr C_2$ | \( 1 - 3356 T^{2} + 733558 p^{2} T^{4} - 3356 p^{14} T^{6} + p^{28} T^{8} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 + 509140 T^{2} + 72934926198 T^{4} + 509140 p^{14} T^{6} + p^{28} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 120 p T + 16291542 T^{2} + 120 p^{8} T^{3} + p^{14} T^{4} )^{2} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 + 41232044 T^{2} + 4743941185389462 T^{4} + 41232044 p^{14} T^{6} + p^{28} T^{8} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 - 1097958084 T^{2} + 599559224884126022 T^{4} - 1097958084 p^{14} T^{6} + p^{28} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 18168 T + 1662196934 T^{2} + 18168 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 - 6516361260 T^{2} + 29623716854444200118 T^{4} - 6516361260 p^{14} T^{6} + p^{28} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 240948 T + 48181474894 T^{2} - 240948 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 80672 T + 35845661118 T^{2} + 80672 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 - 149482424180 T^{2} + \)\(12\!\cdots\!78\)\( T^{4} - 149482424180 p^{14} T^{6} + p^{28} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 564732 T + 458552625318 T^{2} - 564732 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 43 | $C_2^2 \wr C_2$ | \( 1 - 1017603527228 T^{2} + \)\(40\!\cdots\!94\)\( T^{4} - 1017603527228 p^{14} T^{6} + p^{28} T^{8} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 - 1347914717964 T^{2} + \)\(90\!\cdots\!62\)\( T^{4} - 1347914717964 p^{14} T^{6} + p^{28} T^{8} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 - 1631951064500 T^{2} + \)\(33\!\cdots\!38\)\( T^{4} - 1631951064500 p^{14} T^{6} + p^{28} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 3873576 T + 8584168792182 T^{2} + 3873576 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 2152564 T + 6779671203966 T^{2} + 2152564 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 - 15225351658460 T^{2} + \)\(13\!\cdots\!58\)\( T^{4} - 15225351658460 p^{14} T^{6} + p^{28} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 603024 T + 11836109777326 T^{2} - 603024 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 - 37443852555236 T^{2} + \)\(58\!\cdots\!42\)\( T^{4} - 37443852555236 p^{14} T^{6} + p^{28} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 2858496 T + 23904359043422 T^{2} + 2858496 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 - 47252838634140 T^{2} + \)\(17\!\cdots\!58\)\( T^{4} - 47252838634140 p^{14} T^{6} + p^{28} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 6820788 T + 97984966472694 T^{2} - 6820788 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 97 | $C_2^2 \wr C_2$ | \( 1 - 52634745659780 T^{2} + \)\(13\!\cdots\!38\)\( T^{4} - 52634745659780 p^{14} T^{6} + p^{28} T^{8} \) |
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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
−7.27953939977539344585865953464, −6.81997088952396495629271835917, −6.70250001979288596012479848763, −6.49426477697060834440584633706, −6.04816109942741081117488574530, −6.04132108466463788194074932319, −5.84013900300406201302545910138, −5.36703973941182060495814854617, −4.73328700614331141422640265741, −4.72179922506369729574661798228, −4.70245587438614294324992508620, −4.53134311182115780571183611633, −3.96433757338193014924644243075, −3.87168894756266547586164736472, −3.18556504731322358699987583689, −3.04886182665135266476237151280, −2.81596196310049407115906911241, −2.49700426135202749860895198186, −2.20104795685041974241562507559, −1.76802851410711618874404985554, −1.38250799636850747783005104187, −1.24302643219373797009655241659, −1.07704577618082192694760157850, −0.43329523480932288996655405591, −0.27558320591004231604622457002,
0.27558320591004231604622457002, 0.43329523480932288996655405591, 1.07704577618082192694760157850, 1.24302643219373797009655241659, 1.38250799636850747783005104187, 1.76802851410711618874404985554, 2.20104795685041974241562507559, 2.49700426135202749860895198186, 2.81596196310049407115906911241, 3.04886182665135266476237151280, 3.18556504731322358699987583689, 3.87168894756266547586164736472, 3.96433757338193014924644243075, 4.53134311182115780571183611633, 4.70245587438614294324992508620, 4.72179922506369729574661798228, 4.73328700614331141422640265741, 5.36703973941182060495814854617, 5.84013900300406201302545910138, 6.04132108466463788194074932319, 6.04816109942741081117488574530, 6.49426477697060834440584633706, 6.70250001979288596012479848763, 6.81997088952396495629271835917, 7.27953939977539344585865953464
