| L(s) = 1 | + 4·3-s − 4·5-s − 7-s + 10·9-s + 3·11-s + 2·13-s − 16·15-s − 9·17-s − 7·19-s − 4·21-s − 9·23-s + 10·25-s + 20·27-s − 3·29-s − 4·31-s + 12·33-s + 4·35-s − 37-s + 8·39-s + 15·41-s + 2·43-s − 40·45-s + 9·47-s + 6·49-s − 36·51-s − 12·53-s − 12·55-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 1.78·5-s − 0.377·7-s + 10/3·9-s + 0.904·11-s + 0.554·13-s − 4.13·15-s − 2.18·17-s − 1.60·19-s − 0.872·21-s − 1.87·23-s + 2·25-s + 3.84·27-s − 0.557·29-s − 0.718·31-s + 2.08·33-s + 0.676·35-s − 0.164·37-s + 1.28·39-s + 2.34·41-s + 0.304·43-s − 5.96·45-s + 1.31·47-s + 6/7·49-s − 5.04·51-s − 1.64·53-s − 1.61·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 67^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 67^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.390437743\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.390437743\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 67 | $C_2^2$ | \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| good | 7 | $D_4\times C_2$ | \( 1 + T - 5 T^{2} - 8 T^{3} - 20 T^{4} - 8 p T^{5} - 5 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 3 T - 7 T^{2} + 18 T^{3} + 36 T^{4} + 18 p T^{5} - 7 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 9 T + 35 T^{2} + 108 T^{3} + 450 T^{4} + 108 p T^{5} + 35 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 7 T + 7 T^{2} + 28 T^{3} + 472 T^{4} + 28 p T^{5} + 7 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + 9 T + p T^{2} )^{2}( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} ) \) |
| 29 | $D_4\times C_2$ | \( 1 + 3 T - 43 T^{2} - 18 T^{3} + 1602 T^{4} - 18 p T^{5} - 43 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 4 T - 17 T^{2} - 116 T^{3} - 368 T^{4} - 116 p T^{5} - 17 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + T + T^{2} - 2 p T^{3} - 38 p T^{4} - 2 p^{2} T^{5} + p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 15 T + 95 T^{2} - 720 T^{3} + 5994 T^{4} - 720 p T^{5} + 95 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - T + 78 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 9 T - 25 T^{2} - 108 T^{3} + 5220 T^{4} - 108 p T^{5} - 25 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 6 T + 82 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 - 6 T + 94 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 + 9 T - 7 T^{2} - 486 T^{3} - 3048 T^{4} - 486 p T^{5} - 7 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 13 T + 96 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 20 T + 175 T^{2} - 1340 T^{3} + 12784 T^{4} - 1340 p T^{5} + 175 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 3 T - 85 T^{2} + 216 T^{3} + 1200 T^{4} + 216 p T^{5} - 85 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - T - 96 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 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
−6.06038684939817165593009062720, −5.86619216306706806704030333286, −5.70356549787210982453563559656, −5.24901008893988001661839511301, −5.06091092285716598876781529798, −4.70735929491790800629947870084, −4.54374952389755975347980614461, −4.45748855632819985049046064803, −4.19053943601754495843404879473, −4.16338147410362468567307596687, −3.83065791111974143272029353699, −3.68301470831653715982920561318, −3.66618168883153304172944837339, −3.33038481974626290022540654007, −3.31116731657755185330221002373, −2.73890376549569602520223501379, −2.54558263575396951309434962677, −2.33962749387423849735500299547, −2.17431635693623432430883163499, −2.14049150307294125063812917251, −1.56953958991253918438757652580, −1.55739000156374333286696310356, −0.950867421841701382995113917762, −0.46343611770326132095497620398, −0.44014533643502251072354119628,
0.44014533643502251072354119628, 0.46343611770326132095497620398, 0.950867421841701382995113917762, 1.55739000156374333286696310356, 1.56953958991253918438757652580, 2.14049150307294125063812917251, 2.17431635693623432430883163499, 2.33962749387423849735500299547, 2.54558263575396951309434962677, 2.73890376549569602520223501379, 3.31116731657755185330221002373, 3.33038481974626290022540654007, 3.66618168883153304172944837339, 3.68301470831653715982920561318, 3.83065791111974143272029353699, 4.16338147410362468567307596687, 4.19053943601754495843404879473, 4.45748855632819985049046064803, 4.54374952389755975347980614461, 4.70735929491790800629947870084, 5.06091092285716598876781529798, 5.24901008893988001661839511301, 5.70356549787210982453563559656, 5.86619216306706806704030333286, 6.06038684939817165593009062720
