| L(s) = 1 | + 4·4-s − 4·7-s + 7·16-s − 16·28-s − 8·37-s − 8·43-s − 24·47-s − 2·49-s + 24·59-s + 8·64-s + 16·67-s − 16·79-s + 24·83-s + 24·89-s + 32·109-s − 28·112-s + 16·121-s + 127-s + 131-s + 137-s + 139-s − 32·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 2·4-s − 1.51·7-s + 7/4·16-s − 3.02·28-s − 1.31·37-s − 1.21·43-s − 3.50·47-s − 2/7·49-s + 3.12·59-s + 64-s + 1.95·67-s − 1.80·79-s + 2.63·83-s + 2.54·89-s + 3.06·109-s − 2.64·112-s + 1.45·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.63·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.458096730\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.458096730\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - p^{2} T^{2} + 9 T^{4} - p^{4} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 114 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 4 T^{2} - 90 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 28 T^{2} + 486 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 2934 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 88 T^{2} + 5826 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 12 T + 142 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 1206 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 + 32 T^{2} + 78 p T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 244 T^{2} + 25110 T^{4} - 244 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 12 T + 190 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 15606 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.71448721601969620544894681179, −6.61819148397085737027265794916, −6.28424321447028887858397513100, −6.14516664035955040737070642987, −6.10090280884624670090084308070, −5.72641893913960476278511251009, −5.57248216584163989013969634864, −5.01789473856834667069167758970, −4.97237481236319717081153515247, −4.90145846275400143737148333785, −4.77023597517994443658583942541, −4.08831125736766266272538288536, −3.85619590074714188386106387303, −3.63244661424104735421847718263, −3.42709329611219382762092757245, −3.35697725953070575328903991881, −3.03936652351935202388385817938, −2.66505240187836457200225993604, −2.59766341470026924219244363866, −2.04041108916658314969530758287, −1.97788079613858379426440910948, −1.73884013747167011942458183318, −1.40445548162736190775702066392, −0.60692318059298194346119915080, −0.49742486219028495809722545138,
0.49742486219028495809722545138, 0.60692318059298194346119915080, 1.40445548162736190775702066392, 1.73884013747167011942458183318, 1.97788079613858379426440910948, 2.04041108916658314969530758287, 2.59766341470026924219244363866, 2.66505240187836457200225993604, 3.03936652351935202388385817938, 3.35697725953070575328903991881, 3.42709329611219382762092757245, 3.63244661424104735421847718263, 3.85619590074714188386106387303, 4.08831125736766266272538288536, 4.77023597517994443658583942541, 4.90145846275400143737148333785, 4.97237481236319717081153515247, 5.01789473856834667069167758970, 5.57248216584163989013969634864, 5.72641893913960476278511251009, 6.10090280884624670090084308070, 6.14516664035955040737070642987, 6.28424321447028887858397513100, 6.61819148397085737027265794916, 6.71448721601969620544894681179
