| L(s) = 1 | + 2·3-s − 2·5-s + 5·9-s + 2·11-s − 20·13-s − 4·15-s + 10·17-s − 4·19-s − 4·23-s + 25-s + 10·27-s + 4·29-s + 12·31-s + 4·33-s + 4·37-s − 40·39-s − 8·41-s + 24·43-s − 10·45-s + 2·47-s + 20·51-s + 16·53-s − 4·55-s − 8·57-s + 4·59-s + 16·61-s + 40·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 5/3·9-s + 0.603·11-s − 5.54·13-s − 1.03·15-s + 2.42·17-s − 0.917·19-s − 0.834·23-s + 1/5·25-s + 1.92·27-s + 0.742·29-s + 2.15·31-s + 0.696·33-s + 0.657·37-s − 6.40·39-s − 1.24·41-s + 3.65·43-s − 1.49·45-s + 0.291·47-s + 2.80·51-s + 2.19·53-s − 0.539·55-s − 1.05·57-s + 0.520·59-s + 2.04·61-s + 4.96·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.653661457\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.653661457\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 2 T - T^{2} + 2 T^{3} + 4 T^{4} + 2 p T^{5} - p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 2 T - p T^{2} + 14 T^{3} + 60 T^{4} + 14 p T^{5} - p^{3} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 10 T + 49 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 10 T + 43 T^{2} - 230 T^{3} + 1260 T^{4} - 230 p T^{5} + 43 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 4 T + 6 T^{2} - 112 T^{3} - 565 T^{4} - 112 p T^{5} + 6 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 4 T - 32 T^{2} + 8 T^{3} + 1407 T^{4} + 8 p T^{5} - 32 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 12 T + 48 T^{2} - 408 T^{3} + 3791 T^{4} - 408 p T^{5} + 48 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 4 T - 60 T^{2} - 8 T^{3} + 3815 T^{4} - 8 p T^{5} - 60 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 4 T + 84 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 12 T + 90 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 2 T + 7 T^{2} + 194 T^{3} - 2388 T^{4} + 194 p T^{5} + 7 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 16 T + 104 T^{2} - 736 T^{3} + 6727 T^{4} - 736 p T^{5} + 104 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 4 T - 104 T^{2} - 8 T^{3} + 9975 T^{4} - 8 p T^{5} - 104 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 16 T + 78 T^{2} - 896 T^{3} + 12347 T^{4} - 896 p T^{5} + 78 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 8 T - 36 T^{2} + 272 T^{3} + 1223 T^{4} + 272 p T^{5} - 36 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 4 T + 74 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 16 T + 54 T^{2} - 896 T^{3} + 16787 T^{4} - 896 p T^{5} + 54 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 2 T + 45 T^{2} + 398 T^{3} - 4876 T^{4} + 398 p T^{5} + 45 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 89 | $C_2^3$ | \( 1 + 110 T^{2} + 4179 T^{4} + 110 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 6 T + 41 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.33129201478935181273469079250, −7.19614249230403352347077863927, −6.82988158071526273027994287343, −6.54865130586743349387444195755, −6.53603596718866712310649130692, −5.93211686092984292947494401388, −5.90979189465625866316598992282, −5.40457195855130201633722106893, −5.16538428132582563149264536619, −4.99102246117350731149441231988, −4.78318275564310309682968900108, −4.66103492921898148795367846600, −4.37209765315206811992515655445, −3.97832525418577425419372863642, −3.84890697726163434164552195334, −3.69273170515887622495671188114, −3.31436461484153477365915187222, −2.70201144718401015257576039169, −2.62632548322809982686739200978, −2.49531079680727329042369027709, −2.18692360294010588135582098552, −2.16953722663812002115986305904, −1.20689452223422705453446108372, −0.876242537426661560987527842652, −0.49710542334889979006308684070,
0.49710542334889979006308684070, 0.876242537426661560987527842652, 1.20689452223422705453446108372, 2.16953722663812002115986305904, 2.18692360294010588135582098552, 2.49531079680727329042369027709, 2.62632548322809982686739200978, 2.70201144718401015257576039169, 3.31436461484153477365915187222, 3.69273170515887622495671188114, 3.84890697726163434164552195334, 3.97832525418577425419372863642, 4.37209765315206811992515655445, 4.66103492921898148795367846600, 4.78318275564310309682968900108, 4.99102246117350731149441231988, 5.16538428132582563149264536619, 5.40457195855130201633722106893, 5.90979189465625866316598992282, 5.93211686092984292947494401388, 6.53603596718866712310649130692, 6.54865130586743349387444195755, 6.82988158071526273027994287343, 7.19614249230403352347077863927, 7.33129201478935181273469079250
