| L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s − 2·7-s + 2·8-s + 7·9-s + 4·11-s + 2·12-s + 2·14-s + 8·17-s − 7·18-s + 4·19-s − 4·21-s − 4·22-s − 6·23-s + 4·24-s + 22·27-s − 2·28-s − 2·29-s − 16·31-s + 4·32-s + 8·33-s − 8·34-s + 7·36-s − 4·38-s − 6·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.755·7-s + 0.707·8-s + 7/3·9-s + 1.20·11-s + 0.577·12-s + 0.534·14-s + 1.94·17-s − 1.64·18-s + 0.917·19-s − 0.872·21-s − 0.852·22-s − 1.25·23-s + 0.816·24-s + 4.23·27-s − 0.377·28-s − 0.371·29-s − 2.87·31-s + 0.707·32-s + 1.39·33-s − 1.37·34-s + 7/6·36-s − 0.648·38-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.389335568\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.389335568\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 + T - 3 T^{3} - 5 T^{4} - 3 p T^{5} + p^{3} T^{7} + p^{4} T^{8} \) |
| 3 | $C_2^2$ | \( ( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 4 T + 3 T^{2} + 36 T^{3} - 128 T^{4} + 36 p T^{5} + 3 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 8 T + 27 T^{2} - 24 T^{3} - 8 T^{4} - 24 p T^{5} + 27 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 4 T - 13 T^{2} + 36 T^{3} + 176 T^{4} + 36 p T^{5} - 13 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 + 2 T - 3 T^{2} - 102 T^{3} - 908 T^{4} - 102 p T^{5} - 3 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 - 61 T^{2} + 2352 T^{4} - 61 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 6 T - 7 T^{2} - 6 p T^{3} - 36 p T^{4} - 6 p^{2} T^{5} - 7 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - T^{2} - 3480 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 12 T + 83 T^{2} + 972 T^{3} - 11544 T^{4} + 972 p T^{5} + 83 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 16 T + 158 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 4 T + 110 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 2 T - 123 T^{2} - 102 T^{3} + 7852 T^{4} - 102 p T^{5} - 123 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 24 T + 251 T^{2} + 3144 T^{3} + 40344 T^{4} + 3144 p T^{5} + 251 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.510092599688663820859761106275, −8.034059161771232588227221268947, −7.972606871790070414945549898896, −7.66758887332794813686196299996, −7.54827420226115694303041201209, −7.33900165743966071498081908284, −6.92339586144494068228484129047, −6.74359262157074900666196594213, −6.59410422657347910222191309214, −6.26732699411462469006294834761, −5.98161892275181840749330747399, −5.44493730143086459700876841250, −5.20896297522348838823106162448, −5.00229982813411728945469653823, −4.73553918528322940323808604370, −3.98071201890444720609780830678, −3.92016790893572744347527375906, −3.76875283990016121837267682619, −3.50328552729800497329354268140, −2.91937716854347224178348248063, −2.85810061674438127217345055954, −2.05415145370801478712475317421, −1.55780284555723357196890941138, −1.52332187916338461010133311046, −0.977804594119646731272313460937,
0.977804594119646731272313460937, 1.52332187916338461010133311046, 1.55780284555723357196890941138, 2.05415145370801478712475317421, 2.85810061674438127217345055954, 2.91937716854347224178348248063, 3.50328552729800497329354268140, 3.76875283990016121837267682619, 3.92016790893572744347527375906, 3.98071201890444720609780830678, 4.73553918528322940323808604370, 5.00229982813411728945469653823, 5.20896297522348838823106162448, 5.44493730143086459700876841250, 5.98161892275181840749330747399, 6.26732699411462469006294834761, 6.59410422657347910222191309214, 6.74359262157074900666196594213, 6.92339586144494068228484129047, 7.33900165743966071498081908284, 7.54827420226115694303041201209, 7.66758887332794813686196299996, 7.972606871790070414945549898896, 8.034059161771232588227221268947, 8.510092599688663820859761106275
