| L(s) = 1 | + 5·2-s − 30·3-s + 4·4-s − 25·5-s − 150·6-s + 150·7-s − 90·8-s + 341·9-s − 125·10-s + 11·11-s − 120·12-s − 90·13-s + 750·14-s + 750·15-s − 580·16-s + 70·17-s + 1.70e3·18-s − 825·19-s − 100·20-s − 4.50e3·21-s + 55·22-s − 280·23-s + 2.70e3·24-s + 250·25-s − 450·26-s − 1.04e3·27-s + 600·28-s + ⋯ |
| L(s) = 1 | + 5/4·2-s − 3.33·3-s + 1/4·4-s − 5-s − 4.16·6-s + 3.06·7-s − 1.40·8-s + 4.20·9-s − 5/4·10-s + 1/11·11-s − 5/6·12-s − 0.532·13-s + 3.82·14-s + 10/3·15-s − 2.26·16-s + 0.242·17-s + 5.26·18-s − 2.28·19-s − 1/4·20-s − 10.2·21-s + 5/44·22-s − 0.529·23-s + 4.68·24-s + 2/5·25-s − 0.665·26-s − 1.42·27-s + 0.765·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.5385222077\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5385222077\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_4$ | \( 1 + p^{2} T + 3 p^{3} T^{2} + p^{5} T^{3} + p^{6} T^{4} \) |
| 11 | $C_4$ | \( 1 - p T - 19 p^{3} T^{2} - p^{5} T^{3} + p^{8} T^{4} \) |
| good | 2 | $C_2^2:C_4$ | \( 1 - 5 T + 21 T^{2} + 5 T^{3} + 21 T^{4} + 5 p^{4} T^{5} + 21 p^{8} T^{6} - 5 p^{12} T^{7} + p^{16} T^{8} \) |
| 3 | $C_2^2:C_4$ | \( 1 + 10 p T + 559 T^{2} + 7580 T^{3} + 77341 T^{4} + 7580 p^{4} T^{5} + 559 p^{8} T^{6} + 10 p^{13} T^{7} + p^{16} T^{8} \) |
| 7 | $C_2^2:C_4$ | \( 1 - 150 T + 13501 T^{2} - 944100 T^{3} + 51506101 T^{4} - 944100 p^{4} T^{5} + 13501 p^{8} T^{6} - 150 p^{12} T^{7} + p^{16} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 + 90 T + p^{4} T^{2} - 90 p^{4} T^{3} + 366164521 T^{4} - 90 p^{8} T^{5} + p^{12} T^{6} + 90 p^{12} T^{7} + p^{16} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 - 70 T + 102491 T^{2} - 32188380 T^{3} + 3537136801 T^{4} - 32188380 p^{4} T^{5} + 102491 p^{8} T^{6} - 70 p^{12} T^{7} + p^{16} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 + 825 T + 235921 T^{2} - 54131145 T^{3} - 49994342324 T^{4} - 54131145 p^{4} T^{5} + 235921 p^{8} T^{6} + 825 p^{12} T^{7} + p^{16} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 140 T + 348262 T^{2} + 140 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 29 | $C_2^2:C_4$ | \( 1 - 590 T + 1306721 T^{2} - 623316810 T^{3} + 1011351952681 T^{4} - 623316810 p^{4} T^{5} + 1306721 p^{8} T^{6} - 590 p^{12} T^{7} + p^{16} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 + 838 T - 598557 T^{2} - 163917844 T^{3} + 761195826725 T^{4} - 163917844 p^{4} T^{5} - 598557 p^{8} T^{6} + 838 p^{12} T^{7} + p^{16} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 + 20 T - 1461921 T^{2} - 1712003630 T^{3} + 3115391734301 T^{4} - 1712003630 p^{4} T^{5} - 1461921 p^{8} T^{6} + 20 p^{12} T^{7} + p^{16} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 - 6520 T + 20827481 T^{2} - 46491013680 T^{3} + 84585958558081 T^{4} - 46491013680 p^{4} T^{5} + 20827481 p^{8} T^{6} - 6520 p^{12} T^{7} + p^{16} T^{8} \) |
| 43 | $C_2^2:C_4$ | \( 1 - 11339399 T^{2} + 55518339064201 T^{4} - 11339399 p^{8} T^{6} + p^{16} T^{8} \) |
| 47 | $C_2^2:C_4$ | \( 1 - 1740 T - 3009921 T^{2} + 11383493860 T^{3} - 4230245045919 T^{4} + 11383493860 p^{4} T^{5} - 3009921 p^{8} T^{6} - 1740 p^{12} T^{7} + p^{16} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 + 2590 T - 2154381 T^{2} - 25224520180 T^{3} - 47897378326939 T^{4} - 25224520180 p^{4} T^{5} - 2154381 p^{8} T^{6} + 2590 p^{12} T^{7} + p^{16} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 + 7883 T + 12451353 T^{2} - 95407920539 T^{3} - 594621428279620 T^{4} - 95407920539 p^{4} T^{5} + 12451353 p^{8} T^{6} + 7883 p^{12} T^{7} + p^{16} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 - 10060 T + 42999721 T^{2} - 68795890070 T^{3} + 38399337155581 T^{4} - 68795890070 p^{4} T^{5} + 42999721 p^{8} T^{6} - 10060 p^{12} T^{7} + p^{16} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 10875 T + 64361397 T^{2} + 10875 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 71 | $C_2^2:C_4$ | \( 1 - 5018 T - 8529997 T^{2} + 155950068644 T^{3} - 533891359186235 T^{4} + 155950068644 p^{4} T^{5} - 8529997 p^{8} T^{6} - 5018 p^{12} T^{7} + p^{16} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 - 15970 T + 121008271 T^{2} - 601077242780 T^{3} + 2903672078002921 T^{4} - 601077242780 p^{4} T^{5} + 121008271 p^{8} T^{6} - 15970 p^{12} T^{7} + p^{16} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 - 10280 T + 90514561 T^{2} - 708596282320 T^{3} + 4591087445284801 T^{4} - 708596282320 p^{4} T^{5} + 90514561 p^{8} T^{6} - 10280 p^{12} T^{7} + p^{16} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 + 9595 T + 22760791 T^{2} - 426051293695 T^{3} - 5066857677832424 T^{4} - 426051293695 p^{4} T^{5} + 22760791 p^{8} T^{6} + 9595 p^{12} T^{7} + p^{16} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 20409 T + 208736841 T^{2} - 20409 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 97 | $C_2^2:C_4$ | \( 1 - 20045 T + 246853029 T^{2} - 3165941526355 T^{3} + 37545154607921156 T^{4} - 3165941526355 p^{4} T^{5} + 246853029 p^{8} T^{6} - 20045 p^{12} T^{7} + p^{16} T^{8} \) |
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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
−10.81002711044276799385533168022, −10.78265652934969844958770514802, −10.70444212047831612194386657166, −9.871863351165217762066000451770, −9.253896813705656387020479546308, −8.957643257270994240236749258437, −8.781622571514527030944146637396, −8.111216854791357931680716932128, −7.998819248484328296588905469486, −7.58160993215090951750160622200, −7.34294229705210099325209786016, −6.45205216199928287660419205099, −6.26714209935703245121205599845, −6.08841632452344175729558634144, −5.77292819937925961246417303087, −5.30510126373816766200091142938, −4.92371924938560679080828556695, −4.80118328788561373688689771611, −4.47169844153118975474738770709, −4.21362345248375995354667118813, −3.55343675949797438368704517216, −2.46019980708459188504068298225, −1.95753226285131325784249930523, −0.76110261474693321413847246838, −0.33722649114550322698986783709,
0.33722649114550322698986783709, 0.76110261474693321413847246838, 1.95753226285131325784249930523, 2.46019980708459188504068298225, 3.55343675949797438368704517216, 4.21362345248375995354667118813, 4.47169844153118975474738770709, 4.80118328788561373688689771611, 4.92371924938560679080828556695, 5.30510126373816766200091142938, 5.77292819937925961246417303087, 6.08841632452344175729558634144, 6.26714209935703245121205599845, 6.45205216199928287660419205099, 7.34294229705210099325209786016, 7.58160993215090951750160622200, 7.998819248484328296588905469486, 8.111216854791357931680716932128, 8.781622571514527030944146637396, 8.957643257270994240236749258437, 9.253896813705656387020479546308, 9.871863351165217762066000451770, 10.70444212047831612194386657166, 10.78265652934969844958770514802, 10.81002711044276799385533168022
