| L(s) = 1 | + 4·5-s + 4·7-s + 8·13-s − 8·17-s + 8·19-s − 8·23-s + 8·25-s + 16·35-s − 8·37-s + 24·43-s + 8·47-s + 8·49-s + 8·53-s − 16·59-s + 8·61-s + 32·65-s − 8·67-s + 12·73-s + 32·79-s + 32·83-s − 32·85-s + 32·91-s + 32·95-s + 12·97-s − 24·101-s − 12·103-s + 24·113-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 1.51·7-s + 2.21·13-s − 1.94·17-s + 1.83·19-s − 1.66·23-s + 8/5·25-s + 2.70·35-s − 1.31·37-s + 3.65·43-s + 1.16·47-s + 8/7·49-s + 1.09·53-s − 2.08·59-s + 1.02·61-s + 3.96·65-s − 0.977·67-s + 1.40·73-s + 3.60·79-s + 3.51·83-s − 3.47·85-s + 3.35·91-s + 3.28·95-s + 1.21·97-s − 2.38·101-s − 1.18·103-s + 2.25·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.94365124\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.94365124\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| good | 7 | $C_2$$\times$$C_2^2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 - 10 T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 104 T^{3} + 322 T^{4} + 104 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 23 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 152 T^{3} + 706 T^{4} + 152 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 96 T^{2} + 3890 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 870 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} - 24 T^{3} - 1582 T^{4} - 24 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} + 424 T^{3} - 4382 T^{4} + 424 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 104 T^{3} - 1262 T^{4} - 104 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 8 T + 80 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 216 T^{3} - 142 T^{4} + 216 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 516 T^{3} + 2798 T^{4} - 516 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 16 T + 198 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 32 T + 512 T^{2} - 6368 T^{3} + 65746 T^{4} - 6368 p T^{5} + 512 p^{2} T^{6} - 32 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.89275401297866063594314921842, −6.26434153043606672627995506095, −6.24577165999011715688046906374, −6.17552251775698841198644251925, −6.12176940237948063943271081238, −5.54756580737387718590019593549, −5.52433586416336050860103365784, −5.46848140792020830002701165635, −5.13347460508009547059651137968, −4.74836876970103008867975065686, −4.66176288548161590364313911052, −4.27313689992625560565751424729, −4.20541886378882295497757205288, −3.93996407675402532049898225894, −3.38998894656547614869955172195, −3.37137859238186084795665716364, −3.33479139889774377640056037894, −2.48783365739731674744688644026, −2.24921700038443239087836542337, −2.16931938709952983728979493280, −2.14849250215173508914196872774, −1.58749387957120286401259056824, −1.29664129240195812460883137897, −0.889145791518143419807861616668, −0.69983923757089641221210290102,
0.69983923757089641221210290102, 0.889145791518143419807861616668, 1.29664129240195812460883137897, 1.58749387957120286401259056824, 2.14849250215173508914196872774, 2.16931938709952983728979493280, 2.24921700038443239087836542337, 2.48783365739731674744688644026, 3.33479139889774377640056037894, 3.37137859238186084795665716364, 3.38998894656547614869955172195, 3.93996407675402532049898225894, 4.20541886378882295497757205288, 4.27313689992625560565751424729, 4.66176288548161590364313911052, 4.74836876970103008867975065686, 5.13347460508009547059651137968, 5.46848140792020830002701165635, 5.52433586416336050860103365784, 5.54756580737387718590019593549, 6.12176940237948063943271081238, 6.17552251775698841198644251925, 6.24577165999011715688046906374, 6.26434153043606672627995506095, 6.89275401297866063594314921842
