L(s) = 1 | + 2·5-s + 2·7-s − 2·11-s + 2·13-s + 16·19-s + 6·23-s + 11·25-s − 10·29-s + 10·31-s + 4·35-s − 16·37-s − 14·41-s − 10·43-s − 2·47-s + 9·49-s + 16·53-s − 4·55-s + 14·59-s − 6·61-s + 4·65-s − 10·67-s − 8·71-s − 4·77-s + 22·79-s − 6·83-s + 32·89-s + 4·91-s + ⋯ |
L(s) = 1 | + 0.894·5-s + 0.755·7-s − 0.603·11-s + 0.554·13-s + 3.67·19-s + 1.25·23-s + 11/5·25-s − 1.85·29-s + 1.79·31-s + 0.676·35-s − 2.63·37-s − 2.18·41-s − 1.52·43-s − 0.291·47-s + 9/7·49-s + 2.19·53-s − 0.539·55-s + 1.82·59-s − 0.768·61-s + 0.496·65-s − 1.22·67-s − 0.949·71-s − 0.455·77-s + 2.47·79-s − 0.658·83-s + 3.39·89-s + 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.207950510\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.207950510\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
good | 5 | $C_2^2$ | \( ( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 2 T - 5 T^{2} + 10 T^{3} + 4 T^{4} + 10 p T^{5} - 5 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 2 T - 13 T^{2} - 10 T^{3} + 124 T^{4} - 10 p T^{5} - 13 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 2 T + T^{2} + 46 T^{3} - 212 T^{4} + 46 p T^{5} + p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 23 | $D_4\times C_2$ | \( 1 - 6 T - 13 T^{2} - 18 T^{3} + 1044 T^{4} - 18 p T^{5} - 13 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 10 T + 41 T^{2} + 10 T^{3} - 260 T^{4} + 10 p T^{5} + 41 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 10 T + 19 T^{2} - 190 T^{3} + 2500 T^{4} - 190 p T^{5} + 19 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 14 T + 89 T^{2} + 350 T^{3} + 1732 T^{4} + 350 p T^{5} + 89 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 + 10 T + p T^{2} )^{2}( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} ) \) |
| 47 | $D_4\times C_2$ | \( 1 + 2 T - 37 T^{2} - 106 T^{3} - 716 T^{4} - 106 p T^{5} - 37 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 8 T + 98 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 14 T + 83 T^{2} + 70 T^{3} - 2276 T^{4} + 70 p T^{5} + 83 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 6 T - 71 T^{2} - 90 T^{3} + 5532 T^{4} - 90 p T^{5} - 71 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 10 T - 5 T^{2} - 290 T^{3} - 164 T^{4} - 290 p T^{5} - 5 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 4 T + 50 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 22 T + 211 T^{2} - 2530 T^{3} + 30052 T^{4} - 2530 p T^{5} + 211 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 6 T - 133 T^{2} + 18 T^{3} + 18684 T^{4} + 18 p T^{5} - 133 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 16 T + 218 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 2 T - 167 T^{2} - 46 T^{3} + 19444 T^{4} - 46 p T^{5} - 167 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.77664621591988753179092113809, −6.37251928689621295388751361648, −6.27139523363594037707538634087, −5.97361148878903154157960562778, −5.67874807479054002040810889853, −5.31857532502498323885209081268, −5.24634318619143262840983201375, −5.16626778037866505120404592816, −5.14450345167735074413660997233, −4.93418879114469726116586135294, −4.58423533164180212757894368288, −4.21386504088834079915415819672, −3.96865793867901402036772110989, −3.55066214829680230130596966838, −3.31052914307582944325053660132, −3.23574429797695283510368643474, −3.16850317437070569483885122609, −2.82466536515815074244255905129, −2.29480020105080556231685410458, −2.13356210915103850722693014870, −1.87455573377407387250819280943, −1.45016841395675930628219374714, −1.07336079612621537608805950955, −1.04951655921322591573620054157, −0.49534822018451821374829266810,
0.49534822018451821374829266810, 1.04951655921322591573620054157, 1.07336079612621537608805950955, 1.45016841395675930628219374714, 1.87455573377407387250819280943, 2.13356210915103850722693014870, 2.29480020105080556231685410458, 2.82466536515815074244255905129, 3.16850317437070569483885122609, 3.23574429797695283510368643474, 3.31052914307582944325053660132, 3.55066214829680230130596966838, 3.96865793867901402036772110989, 4.21386504088834079915415819672, 4.58423533164180212757894368288, 4.93418879114469726116586135294, 5.14450345167735074413660997233, 5.16626778037866505120404592816, 5.24634318619143262840983201375, 5.31857532502498323885209081268, 5.67874807479054002040810889853, 5.97361148878903154157960562778, 6.27139523363594037707538634087, 6.37251928689621295388751361648, 6.77664621591988753179092113809