| L(s) = 1 | − 4·3-s + 4·5-s − 7-s + 10·9-s + 11-s + 10·13-s − 16·15-s + 12·17-s + 5·19-s + 4·21-s + 23-s + 10·25-s − 20·27-s + 2·29-s − 4·31-s − 4·33-s − 4·35-s + 2·37-s − 40·39-s + 4·41-s − 9·43-s + 40·45-s − 11·49-s − 48·51-s + 13·53-s + 4·55-s − 20·57-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 1.78·5-s − 0.377·7-s + 10/3·9-s + 0.301·11-s + 2.77·13-s − 4.13·15-s + 2.91·17-s + 1.14·19-s + 0.872·21-s + 0.208·23-s + 2·25-s − 3.84·27-s + 0.371·29-s − 0.718·31-s − 0.696·33-s − 0.676·35-s + 0.328·37-s − 6.40·39-s + 0.624·41-s − 1.37·43-s + 5.96·45-s − 1.57·49-s − 6.72·51-s + 1.78·53-s + 0.539·55-s − 2.64·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.573574092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.573574092\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 5 | $C_1$ | \( ( 1 - T )^{4} \) |
| 31 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 7 | $C_2 \wr S_4$ | \( 1 + T + 12 T^{2} - 9 T^{3} + 72 T^{4} - 9 p T^{5} + 12 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^3:S_4$ | \( 1 - T + 18 T^{2} + p T^{3} + 210 T^{4} + p^{2} T^{5} + 18 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 10 T + 80 T^{2} - 404 T^{3} + 1730 T^{4} - 404 p T^{5} + 80 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 12 T + 92 T^{2} - 512 T^{3} + 2326 T^{4} - 512 p T^{5} + 92 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 5 T + 60 T^{2} - 269 T^{3} + 1590 T^{4} - 269 p T^{5} + 60 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - T + 70 T^{2} - 13 T^{3} + 2126 T^{4} - 13 p T^{5} + 70 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 2 T + 50 T^{2} + 128 T^{3} + 882 T^{4} + 128 p T^{5} + 50 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 2 T + 140 T^{2} - 212 T^{3} + 7630 T^{4} - 212 p T^{5} + 140 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 4 T + 76 T^{2} - 316 T^{3} + 4918 T^{4} - 316 p T^{5} + 76 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 9 T + 102 T^{2} + 749 T^{3} + 6938 T^{4} + 749 p T^{5} + 102 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 88 T^{2} + 340 T^{3} + 4022 T^{4} + 340 p T^{5} + 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 13 T + 100 T^{2} - 907 T^{3} + 8954 T^{4} - 907 p T^{5} + 100 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 6 T + 226 T^{2} + 1004 T^{3} + 19790 T^{4} + 1004 p T^{5} + 226 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 92 T^{2} - 64 T^{3} + 6550 T^{4} - 64 p T^{5} + 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 8 T + 144 T^{2} + 1346 T^{3} + 13306 T^{4} + 1346 p T^{5} + 144 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 9 T + 96 T^{2} - 707 T^{3} - 4836 T^{4} - 707 p T^{5} + 96 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 7 T + 174 T^{2} - 1695 T^{3} + 15380 T^{4} - 1695 p T^{5} + 174 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 5 T + 172 T^{2} - 813 T^{3} + 20122 T^{4} - 813 p T^{5} + 172 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 2 T + 160 T^{2} - 446 T^{3} + 14350 T^{4} - 446 p T^{5} + 160 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 5 T + 162 T^{2} - 275 T^{3} + 11628 T^{4} - 275 p T^{5} + 162 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 6 T + 100 T^{2} - 2 T^{3} + 10534 T^{4} - 2 p T^{5} + 100 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 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
−6.11525592746248333226197009560, −5.72913943451454343100830012656, −5.70688938886268963615589982102, −5.55945302160875514702360991183, −5.39069266793999639002870459164, −5.01675320774817842265578426817, −4.99333545545412449920876010351, −4.84515298845472613607460249866, −4.66713143080423405025287896076, −4.03350153562587154305201785552, −3.97626889780858741850904330414, −3.80792480778094842213132597064, −3.79954432989123038115383474211, −3.10756776765516579937375718606, −3.05716937084526963900372927564, −3.03369429388445219379434785427, −2.97332614378680385962203792116, −2.06187214767197852335627485479, −1.90471728286255752725907180518, −1.77265168212346310202429048159, −1.41533591809715689191173544359, −1.30800025792417976325922498544, −0.907536637095072533516730606718, −0.71690769440226925882490940743, −0.57103281548249474049564989799,
0.57103281548249474049564989799, 0.71690769440226925882490940743, 0.907536637095072533516730606718, 1.30800025792417976325922498544, 1.41533591809715689191173544359, 1.77265168212346310202429048159, 1.90471728286255752725907180518, 2.06187214767197852335627485479, 2.97332614378680385962203792116, 3.03369429388445219379434785427, 3.05716937084526963900372927564, 3.10756776765516579937375718606, 3.79954432989123038115383474211, 3.80792480778094842213132597064, 3.97626889780858741850904330414, 4.03350153562587154305201785552, 4.66713143080423405025287896076, 4.84515298845472613607460249866, 4.99333545545412449920876010351, 5.01675320774817842265578426817, 5.39069266793999639002870459164, 5.55945302160875514702360991183, 5.70688938886268963615589982102, 5.72913943451454343100830012656, 6.11525592746248333226197009560
