| L(s) = 1 | − 2·2-s + 4-s + 4·7-s + 2·8-s + 9-s + 12·11-s − 8·14-s − 4·16-s − 2·18-s − 12·19-s − 24·22-s − 6·23-s + 4·28-s + 4·29-s + 2·32-s + 36-s + 8·37-s + 24·38-s + 24·43-s + 12·44-s + 12·46-s − 28·47-s + 18·49-s + 8·56-s − 8·58-s − 12·59-s + 4·63-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s + 1.51·7-s + 0.707·8-s + 1/3·9-s + 3.61·11-s − 2.13·14-s − 16-s − 0.471·18-s − 2.75·19-s − 5.11·22-s − 1.25·23-s + 0.755·28-s + 0.742·29-s + 0.353·32-s + 1/6·36-s + 1.31·37-s + 3.89·38-s + 3.65·43-s + 1.80·44-s + 1.76·46-s − 4.08·47-s + 18/7·49-s + 1.06·56-s − 1.05·58-s − 1.56·59-s + 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.02494316093\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02494316093\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 12 T + 73 T^{2} - 300 T^{3} + 1032 T^{4} - 300 p T^{5} + 73 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^3$ | \( 1 + 18 T^{2} + 35 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 12 T + 82 T^{2} + 408 T^{3} + 1707 T^{4} + 408 p T^{5} + 82 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 6 T + 57 T^{2} + 270 T^{3} + 1772 T^{4} + 270 p T^{5} + 57 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 4 T - 43 T^{2} - 4 T^{3} + 2176 T^{4} - 4 p T^{5} - 43 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 8 T + T^{2} + 88 T^{3} + 232 T^{4} + 88 p T^{5} + p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^3$ | \( 1 + 78 T^{2} + 4403 T^{4} + 78 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 24 T + 7 p T^{2} - 2616 T^{3} + 18288 T^{4} - 2616 p T^{5} + 7 p^{3} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 14 T + 131 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 44 T^{2} - 810 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 12 T + 153 T^{2} + 1260 T^{3} + 10376 T^{4} + 1260 p T^{5} + 153 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^3$ | \( 1 - 14 T^{2} - 3525 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 16 T + 70 T^{2} + 832 T^{3} + 12955 T^{4} + 832 p T^{5} + 70 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 36 T + 678 T^{2} + 8856 T^{3} + 86147 T^{4} + 8856 p T^{5} + 678 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 79 | $D_{4}$ | \( ( 1 - 14 T + 159 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 + 4 T + 122 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 12 T + 222 T^{2} + 2088 T^{3} + 26627 T^{4} + 2088 p T^{5} + 222 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 8 T - 134 T^{2} + 32 T^{3} + 23587 T^{4} + 32 p T^{5} - 134 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.52780036872982393880822504667, −6.47385069445989260620476610019, −6.22078502000088631904605058969, −6.08477507264267854478374263114, −5.80326963845925442714195196531, −5.65223315395042425828216020078, −5.12775587547705814606311775282, −5.03124601971198697193202645539, −4.61609976334625588737203072565, −4.47735190577552411028307541481, −4.27377350130464602063372018611, −4.21647730580915101101002468683, −4.03281687600474836194425551405, −3.81801583348656737454018452627, −3.68403487159781515898243993618, −3.07244799578369672266450081523, −2.68959371362839597099589455204, −2.55810700681985800882514349413, −2.33505684365482430501948004488, −1.65418911833453985795087634164, −1.63091603190079144836965393412, −1.39324590336104368282217687876, −1.34162297352635615979731772493, −0.919269934217965818059530015140, −0.03482419007793967399000756086,
0.03482419007793967399000756086, 0.919269934217965818059530015140, 1.34162297352635615979731772493, 1.39324590336104368282217687876, 1.63091603190079144836965393412, 1.65418911833453985795087634164, 2.33505684365482430501948004488, 2.55810700681985800882514349413, 2.68959371362839597099589455204, 3.07244799578369672266450081523, 3.68403487159781515898243993618, 3.81801583348656737454018452627, 4.03281687600474836194425551405, 4.21647730580915101101002468683, 4.27377350130464602063372018611, 4.47735190577552411028307541481, 4.61609976334625588737203072565, 5.03124601971198697193202645539, 5.12775587547705814606311775282, 5.65223315395042425828216020078, 5.80326963845925442714195196531, 6.08477507264267854478374263114, 6.22078502000088631904605058969, 6.47385069445989260620476610019, 6.52780036872982393880822504667
