| L(s) = 1 | + 2·3-s + 4-s + 6·7-s + 9-s − 6·11-s + 2·12-s + 6·19-s + 12·21-s − 6·23-s + 14·25-s − 2·27-s + 6·28-s + 6·29-s − 12·33-s + 36-s − 12·41-s + 2·43-s − 6·44-s + 16·49-s + 12·53-s + 12·57-s + 48·59-s − 20·61-s + 6·63-s − 64-s + 6·67-s − 12·69-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/2·4-s + 2.26·7-s + 1/3·9-s − 1.80·11-s + 0.577·12-s + 1.37·19-s + 2.61·21-s − 1.25·23-s + 14/5·25-s − 0.384·27-s + 1.13·28-s + 1.11·29-s − 2.08·33-s + 1/6·36-s − 1.87·41-s + 0.304·43-s − 0.904·44-s + 16/7·49-s + 1.64·53-s + 1.58·57-s + 6.24·59-s − 2.56·61-s + 0.755·63-s − 1/8·64-s + 0.733·67-s − 1.44·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.172608172\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.172608172\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 13 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 6 T + 20 T^{2} - 48 T^{3} + 99 T^{4} - 48 p T^{5} + 20 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 6 T + 28 T^{2} + 96 T^{3} + 267 T^{4} + 96 p T^{5} + 28 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^3$ | \( 1 - 7 T^{2} - 240 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 6 T + 44 T^{2} - 192 T^{3} + 891 T^{4} - 192 p T^{5} + 44 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 6 T + 8 T^{2} - 108 T^{3} - 573 T^{4} - 108 p T^{5} + 8 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 28 T^{2} + 390 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^3$ | \( 1 + 65 T^{2} + 2856 T^{4} + 65 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 12 T + 133 T^{2} + 1020 T^{3} + 7512 T^{4} + 1020 p T^{5} + 133 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 2 T - 56 T^{2} + 52 T^{3} + 1579 T^{4} + 52 p T^{5} - 56 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 11034 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 - 24 T + 251 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + 20 T + 205 T^{2} + 1460 T^{3} + 9904 T^{4} + 1460 p T^{5} + 205 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 6 T + 68 T^{2} - 336 T^{3} - 549 T^{4} - 336 p T^{5} + 68 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 18 T + 268 T^{2} + 2880 T^{3} + 28227 T^{4} + 2880 p T^{5} + 268 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 17802 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 12 T + 202 T^{2} - 1848 T^{3} + 20067 T^{4} - 1848 p T^{5} + 202 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^3$ | \( 1 + 158 T^{2} + 15555 T^{4} + 158 p^{2} T^{6} + p^{4} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.24169179920622631795885055748, −7.11052537946200296679075513555, −6.73248325221670067147457833640, −6.69948417180941829289146079318, −6.13610351757117598306375926365, −6.03550449334635322712855395029, −5.63023292555833975482881065387, −5.35174923056890398699187452260, −5.24355389104594671663375440978, −5.08618330964465362379215234599, −5.04416603740681225376083091132, −4.42946382549419872590325221517, −4.35595766257429290147639393346, −4.29207659411853828844140492129, −3.60635131157234954682770107615, −3.56045463182859407359582025941, −3.08152972642832159423708186833, −2.96394436202217255166345165232, −2.53338170433605877357153515962, −2.52352372566081489015974356562, −2.14584683960169396386060310179, −1.71174666854277450570018856280, −1.59371494575164655382235940615, −0.904265926876443744034003981003, −0.69846851254721925888064148043,
0.69846851254721925888064148043, 0.904265926876443744034003981003, 1.59371494575164655382235940615, 1.71174666854277450570018856280, 2.14584683960169396386060310179, 2.52352372566081489015974356562, 2.53338170433605877357153515962, 2.96394436202217255166345165232, 3.08152972642832159423708186833, 3.56045463182859407359582025941, 3.60635131157234954682770107615, 4.29207659411853828844140492129, 4.35595766257429290147639393346, 4.42946382549419872590325221517, 5.04416603740681225376083091132, 5.08618330964465362379215234599, 5.24355389104594671663375440978, 5.35174923056890398699187452260, 5.63023292555833975482881065387, 6.03550449334635322712855395029, 6.13610351757117598306375926365, 6.69948417180941829289146079318, 6.73248325221670067147457833640, 7.11052537946200296679075513555, 7.24169179920622631795885055748
