| L(s) = 1 | − 12·4-s − 5·9-s + 10·11-s + 64·16-s + 98·19-s − 232·29-s − 294·31-s + 60·36-s + 1.40e3·41-s − 120·44-s − 98·49-s − 210·59-s + 826·61-s − 576·64-s − 1.72e3·71-s − 1.17e3·76-s − 206·79-s + 729·81-s − 658·89-s − 50·99-s − 2.75e3·101-s − 2.25e3·109-s + 2.78e3·116-s + 2.68e3·121-s + 3.52e3·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 3/2·4-s − 0.185·9-s + 0.274·11-s + 16-s + 1.18·19-s − 1.48·29-s − 1.70·31-s + 5/18·36-s + 5.33·41-s − 0.411·44-s − 2/7·49-s − 0.463·59-s + 1.73·61-s − 9/8·64-s − 2.88·71-s − 1.77·76-s − 0.293·79-s + 81-s − 0.783·89-s − 0.0507·99-s − 2.71·101-s − 1.97·109-s + 2.22·116-s + 2.01·121-s + 2.55·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3318505031\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3318505031\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 2 p^{2} T^{2} + p^{6} T^{4} \) |
| good | 2 | $C_2^3$ | \( 1 + 3 p^{2} T^{2} + 5 p^{4} T^{4} + 3 p^{8} T^{6} + p^{12} T^{8} \) |
| 3 | $C_2^3$ | \( 1 + 5 T^{2} - 704 T^{4} + 5 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 5 T - 1306 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 4198 T^{2} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 9385 T^{2} + 63940656 T^{4} + 9385 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 49 T - 4458 T^{2} - 49 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 947 T^{2} - 147139080 T^{4} - 947 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + 2 p T + p^{3} T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 147 T - 8182 T^{2} + 147 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 53345 T^{2} + 279962616 T^{4} + 53345 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $C_2$ | \( ( 1 - 350 T + p^{3} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 143638 T^{2} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 67979 T^{2} - 6158070888 T^{4} - 67979 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 205945 T^{2} + 20248981896 T^{4} + 205945 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 105 T - 194354 T^{2} + 105 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 413 T - 56412 T^{2} - 413 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 429301 T^{2} + 93840966432 T^{4} + 429301 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 432 T + p^{3} T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 - 460735 T^{2} + 60942513936 T^{4} - 460735 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 103 T - 482430 T^{2} + 103 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 48890 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 329 T - 596728 T^{2} + 329 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 1047422 T^{2} + p^{6} T^{4} )^{2} \) |
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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
−9.060247483031425653912119698547, −8.824383975292032683366638587331, −8.222491568389599267377798997111, −8.052450119012094090459107370437, −7.59056145309397360110394507302, −7.52741563476860634347429557876, −7.45697162630918669299296306914, −6.95161059913089782740263716462, −6.54670442800657062718079531739, −6.22032648252893880737690125290, −5.77927391683633904544417129589, −5.61748806459738988525750806906, −5.33198388048719196558540267537, −5.27722435464847098587106407376, −4.44089647609380881887977819851, −4.39001246857981053684918402440, −4.11671388402718838197787131014, −3.82842075125766111577848442654, −3.35642694053906931968702762451, −2.92883879202137655784970040231, −2.52857199056945698979615513243, −2.01119215369636447623047367429, −1.23967827047672135735462761624, −1.00320797985453228838312600931, −0.14931801372628997531005733865,
0.14931801372628997531005733865, 1.00320797985453228838312600931, 1.23967827047672135735462761624, 2.01119215369636447623047367429, 2.52857199056945698979615513243, 2.92883879202137655784970040231, 3.35642694053906931968702762451, 3.82842075125766111577848442654, 4.11671388402718838197787131014, 4.39001246857981053684918402440, 4.44089647609380881887977819851, 5.27722435464847098587106407376, 5.33198388048719196558540267537, 5.61748806459738988525750806906, 5.77927391683633904544417129589, 6.22032648252893880737690125290, 6.54670442800657062718079531739, 6.95161059913089782740263716462, 7.45697162630918669299296306914, 7.52741563476860634347429557876, 7.59056145309397360110394507302, 8.052450119012094090459107370437, 8.222491568389599267377798997111, 8.824383975292032683366638587331, 9.060247483031425653912119698547
