| L(s) = 1 | − 4·5-s − 4·7-s − 2·9-s + 4·11-s + 8·13-s − 8·16-s − 8·17-s + 8·19-s + 8·25-s + 20·29-s − 16·31-s + 16·35-s − 8·37-s − 16·41-s + 8·45-s − 20·47-s + 8·49-s − 4·53-s − 16·55-s − 8·59-s + 8·63-s − 32·65-s + 24·71-s + 16·73-s − 16·77-s − 20·79-s + 32·80-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 1.51·7-s − 2/3·9-s + 1.20·11-s + 2.21·13-s − 2·16-s − 1.94·17-s + 1.83·19-s + 8/5·25-s + 3.71·29-s − 2.87·31-s + 2.70·35-s − 1.31·37-s − 2.49·41-s + 1.19·45-s − 2.91·47-s + 8/7·49-s − 0.549·53-s − 2.15·55-s − 1.04·59-s + 1.00·63-s − 3.96·65-s + 2.84·71-s + 1.87·73-s − 1.82·77-s − 2.25·79-s + 3.57·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2765438637\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2765438637\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2} \) |
| 5 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 8 T^{3} - T^{4} + 8 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 - 18 T^{2} + p^{2} T^{4} ) \) |
| 17 | $D_{4}$ | \( ( 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 176 T^{3} + 959 T^{4} - 176 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 70 T^{2} + 2243 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 10 T + 73 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 928 T^{3} + 5999 T^{4} + 928 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} - 280 T^{3} - 2734 T^{4} - 280 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 848 T^{3} + 5474 T^{4} + 848 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 90 T^{2} + 5563 T^{4} - 90 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 20 T + 200 T^{2} + 1840 T^{3} + 14903 T^{4} + 1840 p T^{5} + 200 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 2 T + 97 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} - 104 T^{3} - 4846 T^{4} - 104 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 112 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - 6062 T^{4} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 1600 T^{3} + 19271 T^{4} - 1600 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 10 T + 143 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 20 T + 200 T^{2} + 2560 T^{3} + 30743 T^{4} + 2560 p T^{5} + 200 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 28 T + 392 T^{2} + 5096 T^{3} + 57599 T^{4} + 5096 p T^{5} + 392 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 800 T^{3} + 19991 T^{4} + 800 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.653160870567070452921989276919, −8.530059442698811856848207744950, −8.317289832908387279792039431917, −7.894150504156955864366333265337, −7.82774656883646789471256226219, −7.13772476495172290886558007442, −6.87908868386057324851750849332, −6.82084429060559135895805656085, −6.71210105696702462375622295506, −6.46344629272368819701451836386, −6.26632381226452616925485420693, −5.83391123504976754757562748370, −5.30752744628342674365248933365, −5.05927466064118876277340027677, −4.85540221144306396035516576361, −4.42305808868078945862088202919, −3.96380462196344232086535265518, −3.92598524329500373726504983588, −3.46691423296353433533872079049, −3.35960043393467267876393293161, −2.90428594755749102083070976201, −2.68963071339564813570180355042, −1.69721431646154716020872007460, −1.45362398774215730435941109604, −0.26936776768805653201568045932,
0.26936776768805653201568045932, 1.45362398774215730435941109604, 1.69721431646154716020872007460, 2.68963071339564813570180355042, 2.90428594755749102083070976201, 3.35960043393467267876393293161, 3.46691423296353433533872079049, 3.92598524329500373726504983588, 3.96380462196344232086535265518, 4.42305808868078945862088202919, 4.85540221144306396035516576361, 5.05927466064118876277340027677, 5.30752744628342674365248933365, 5.83391123504976754757562748370, 6.26632381226452616925485420693, 6.46344629272368819701451836386, 6.71210105696702462375622295506, 6.82084429060559135895805656085, 6.87908868386057324851750849332, 7.13772476495172290886558007442, 7.82774656883646789471256226219, 7.894150504156955864366333265337, 8.317289832908387279792039431917, 8.530059442698811856848207744950, 8.653160870567070452921989276919
