| L(s) = 1 | + 2-s + 4-s + 5-s − 2·9-s + 10-s + 11-s − 2·13-s − 2·18-s + 19-s + 20-s + 22-s + 23-s + 25-s − 2·26-s + 31-s − 32-s − 2·36-s + 38-s + 44-s − 2·45-s + 46-s + 50-s − 2·52-s + 55-s + 62-s − 64-s − 2·65-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s + 5-s − 2·9-s + 10-s + 11-s − 2·13-s − 2·18-s + 19-s + 20-s + 22-s + 23-s + 25-s − 2·26-s + 31-s − 32-s − 2·36-s + 38-s + 44-s − 2·45-s + 46-s + 50-s − 2·52-s + 55-s + 62-s − 64-s − 2·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 79^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 79^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.597427643\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.597427643\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| good | 2 | $C_4\times C_2$ | \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \) |
| 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 5 | $C_4\times C_2$ | \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \) |
| 11 | $C_4\times C_2$ | \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \) |
| 13 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 19 | $C_4\times C_2$ | \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \) |
| 23 | $C_4\times C_2$ | \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 31 | $C_4\times C_2$ | \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_4\times C_2$ | \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_4\times C_2$ | \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \) |
| 83 | $C_1$ | \( ( 1 - T )^{8} \) |
| 89 | $C_4\times C_2$ | \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \) |
| 97 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
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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
−6.31073830660997941292833676860, −5.90171824620450564999641349229, −5.84942517394790515761329877924, −5.56644277039579083052077003886, −5.23377728037433428123523856073, −5.18541923637781256193198440152, −5.07501538190120198681140174816, −4.94998973327039984275824247256, −4.84491414526951794160399749363, −4.35563152644912492675661090504, −4.23456501180263449862779518819, −3.93870697455737173873386735713, −3.59274002488400520036042299138, −3.46423104129680499569693548471, −3.23856142694820928575502665512, −3.07515285098843577488421230566, −2.94681527158056944418307728554, −2.59515710960398674045930546840, −2.30479749010096875475985333883, −2.27543089016608553645157594398, −2.15581877413023965865285525504, −1.62386949582503797664970533366, −1.35545480193398473659176646527, −0.851464515418252297395622892466, −0.62531970869581775371546194955,
0.62531970869581775371546194955, 0.851464515418252297395622892466, 1.35545480193398473659176646527, 1.62386949582503797664970533366, 2.15581877413023965865285525504, 2.27543089016608553645157594398, 2.30479749010096875475985333883, 2.59515710960398674045930546840, 2.94681527158056944418307728554, 3.07515285098843577488421230566, 3.23856142694820928575502665512, 3.46423104129680499569693548471, 3.59274002488400520036042299138, 3.93870697455737173873386735713, 4.23456501180263449862779518819, 4.35563152644912492675661090504, 4.84491414526951794160399749363, 4.94998973327039984275824247256, 5.07501538190120198681140174816, 5.18541923637781256193198440152, 5.23377728037433428123523856073, 5.56644277039579083052077003886, 5.84942517394790515761329877924, 5.90171824620450564999641349229, 6.31073830660997941292833676860
