| L(s) = 1 | − 2-s + 8·5-s + 3·7-s + 3·9-s − 8·10-s − 11-s − 8·13-s − 3·14-s − 11·17-s − 3·18-s − 19-s + 22-s + 18·23-s + 35·25-s + 8·26-s + 6·29-s + 14·31-s + 32-s + 11·34-s + 24·35-s + 2·37-s + 38-s + 4·41-s − 2·43-s + 24·45-s − 18·46-s + 8·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 3.57·5-s + 1.13·7-s + 9-s − 2.52·10-s − 0.301·11-s − 2.21·13-s − 0.801·14-s − 2.66·17-s − 0.707·18-s − 0.229·19-s + 0.213·22-s + 3.75·23-s + 7·25-s + 1.56·26-s + 1.11·29-s + 2.51·31-s + 0.176·32-s + 1.88·34-s + 4.05·35-s + 0.328·37-s + 0.162·38-s + 0.624·41-s − 0.304·43-s + 3.57·45-s − 2.65·46-s + 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 11^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 11^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.846470432\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.846470432\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 11 | $C_4$ | \( 1 + T - 9 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| good | 3 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 5 | $C_2^2:C_4$ | \( 1 - 8 T + 29 T^{2} - 72 T^{3} + 161 T^{4} - 72 p T^{5} + 29 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2^2:C_4$ | \( 1 - 3 T - 3 T^{2} + 5 T^{3} + 36 T^{4} + 5 p T^{5} - 3 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 + 8 T + 51 T^{2} + 244 T^{3} + 1049 T^{4} + 244 p T^{5} + 51 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 + 11 T + 79 T^{2} + 457 T^{3} + 2184 T^{4} + 457 p T^{5} + 79 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 9 T + 55 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2:C_4$ | \( 1 - 6 T - 13 T^{2} + 42 T^{3} + 625 T^{4} + 42 p T^{5} - 13 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 - 14 T + 105 T^{2} - 766 T^{3} + 5129 T^{4} - 766 p T^{5} + 105 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 - 2 T + 27 T^{2} - 10 T^{3} + 641 T^{4} - 10 p T^{5} + 27 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 - 4 T + 55 T^{2} - 236 T^{3} + 3249 T^{4} - 236 p T^{5} + 55 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + T + 85 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 - 8 T + 67 T^{2} - 430 T^{3} + 1071 T^{4} - 430 p T^{5} + 67 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_4\times C_2$ | \( 1 + 2 T - 49 T^{2} - 204 T^{3} + 2189 T^{4} - 204 p T^{5} - 49 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_4\times C_2$ | \( 1 - 10 T + 41 T^{2} + 180 T^{3} - 4219 T^{4} + 180 p T^{5} + 41 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 + 17 T + 53 T^{2} - 461 T^{3} - 4820 T^{4} - 461 p T^{5} + 53 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 71 | $C_2^2:C_4$ | \( 1 + 14 T + 5 T^{2} - 544 T^{3} - 2611 T^{4} - 544 p T^{5} + 5 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 + 6 T + 3 T^{2} + 640 T^{3} + 9141 T^{4} + 640 p T^{5} + 3 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_4\times C_2$ | \( 1 - 12 T + 65 T^{2} + 168 T^{3} - 7151 T^{4} + 168 p T^{5} + 65 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 + 8 T - 49 T^{2} - 6 T^{3} + 7259 T^{4} - 6 p T^{5} - 49 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 6 T + 62 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2:C_4$ | \( 1 - 2 T + 107 T^{2} - 400 T^{3} + 8101 T^{4} - 400 p T^{5} + 107 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.436712610083848133007109235980, −7.51145513266564451104704919384, −7.49627530444988714730757975594, −7.46220679469244254824410716168, −7.32681208293523091011372823298, −6.71881898000411231284983709597, −6.59220376276754030061246984192, −6.44109758146621749695206065768, −6.13108460847911043587454809768, −6.06223837085238675475656327415, −5.35481768084013896823237778449, −5.26546451173855260370565130317, −5.21510612809109219303831221242, −4.57811945031879680456433681279, −4.56037786507942297073302732816, −4.52951266680804488479275254185, −4.32111687798624885735925734803, −3.05709982234276060185523615154, −2.91907774396294469003221831187, −2.66694860294496584681334085296, −2.31451592277990591841759347255, −2.28573191949558345693742648555, −1.52038901077431140202826446830, −1.44079572516520030742125793663, −0.860856831281497763554662376782,
0.860856831281497763554662376782, 1.44079572516520030742125793663, 1.52038901077431140202826446830, 2.28573191949558345693742648555, 2.31451592277990591841759347255, 2.66694860294496584681334085296, 2.91907774396294469003221831187, 3.05709982234276060185523615154, 4.32111687798624885735925734803, 4.52951266680804488479275254185, 4.56037786507942297073302732816, 4.57811945031879680456433681279, 5.21510612809109219303831221242, 5.26546451173855260370565130317, 5.35481768084013896823237778449, 6.06223837085238675475656327415, 6.13108460847911043587454809768, 6.44109758146621749695206065768, 6.59220376276754030061246984192, 6.71881898000411231284983709597, 7.32681208293523091011372823298, 7.46220679469244254824410716168, 7.49627530444988714730757975594, 7.51145513266564451104704919384, 8.436712610083848133007109235980
