| L(s) = 1 | + 3·3-s + 27·5-s + 7-s − 11·11-s + 79·13-s + 81·15-s + 58·17-s + 194·19-s + 3·21-s + 740·23-s + 400·25-s + 62·29-s − 572·31-s − 33·33-s + 27·35-s − 355·37-s + 237·39-s − 53·41-s − 616·43-s + 289·47-s + 163·49-s + 174·51-s − 466·53-s − 297·55-s + 582·57-s − 328·59-s − 645·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 2.41·5-s + 0.0539·7-s − 0.301·11-s + 1.68·13-s + 1.39·15-s + 0.827·17-s + 2.34·19-s + 0.0311·21-s + 6.70·23-s + 16/5·25-s + 0.397·29-s − 3.31·31-s − 0.174·33-s + 0.130·35-s − 1.57·37-s + 0.973·39-s − 0.201·41-s − 2.18·43-s + 0.896·47-s + 0.475·49-s + 0.477·51-s − 1.20·53-s − 0.728·55-s + 1.35·57-s − 0.723·59-s − 1.35·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(11.98678350\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.98678350\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_4$ | \( 1 - p T + p^{2} T^{2} - p^{3} T^{3} + p^{4} T^{4} \) |
| 11 | $C_4$ | \( 1 + p T + 21 p^{2} T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| good | 5 | $C_2^2:C_4$ | \( 1 - 27 T + 329 T^{2} - 4353 T^{3} + 60676 T^{4} - 4353 p^{3} T^{5} + 329 p^{6} T^{6} - 27 p^{9} T^{7} + p^{12} T^{8} \) |
| 7 | $C_2^2:C_4$ | \( 1 - T - 162 T^{2} + 775 p T^{3} + 85661 T^{4} + 775 p^{4} T^{5} - 162 p^{6} T^{6} - p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 - 79 T + 474 T^{2} + 178117 T^{3} - 11302381 T^{4} + 178117 p^{3} T^{5} + 474 p^{6} T^{6} - 79 p^{9} T^{7} + p^{12} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 - 58 T - 3229 T^{2} + 29556 T^{3} + 23834189 T^{4} + 29556 p^{3} T^{5} - 3229 p^{6} T^{6} - 58 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 - 194 T + 27057 T^{2} - 160858 p T^{3} + 832475 p^{2} T^{4} - 160858 p^{4} T^{5} + 27057 p^{6} T^{6} - 194 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $C_2$ | \( ( 1 - 185 T + p^{3} T^{2} )^{4} \) |
| 29 | $C_2^2:C_4$ | \( 1 - 62 T - 3145 T^{2} - 3520392 T^{3} + 792361709 T^{4} - 3520392 p^{3} T^{5} - 3145 p^{6} T^{6} - 62 p^{9} T^{7} + p^{12} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 + 572 T + 136463 T^{2} + 21718114 T^{3} + 3473856175 T^{4} + 21718114 p^{3} T^{5} + 136463 p^{6} T^{6} + 572 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 + 355 T + 39182 T^{2} + 14968355 T^{3} + 5926930819 T^{4} + 14968355 p^{3} T^{5} + 39182 p^{6} T^{6} + 355 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 + 53 T + 608 T^{2} + 15763731 T^{3} + 5049983375 T^{4} + 15763731 p^{3} T^{5} + 608 p^{6} T^{6} + 53 p^{9} T^{7} + p^{12} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 308 T + 71725 T^{2} + 308 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 - 289 T + 38273 T^{2} + 15100935 T^{3} - 7028406964 T^{4} + 15100935 p^{3} T^{5} + 38273 p^{6} T^{6} - 289 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 + 466 T - 51841 T^{2} - 75977048 T^{3} - 21276106951 T^{4} - 75977048 p^{3} T^{5} - 51841 p^{6} T^{6} + 466 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 + 328 T - 114625 T^{2} - 98822102 T^{3} - 8693287141 T^{4} - 98822102 p^{3} T^{5} - 114625 p^{6} T^{6} + 328 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 + 645 T + 312209 T^{2} + 224138295 T^{3} + 155861138116 T^{4} + 224138295 p^{3} T^{5} + 312209 p^{6} T^{6} + 645 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 743 T + 532477 T^{2} + 743 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2:C_4$ | \( 1 + 3 T - 258407 T^{2} - 154164909 T^{3} + 112038345100 T^{4} - 154164909 p^{3} T^{5} - 258407 p^{6} T^{6} + 3 p^{9} T^{7} + p^{12} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 + 1482 T + 442727 T^{2} - 544125990 T^{3} - 606712030919 T^{4} - 544125990 p^{3} T^{5} + 442727 p^{6} T^{6} + 1482 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 + 13 p T - 34710 T^{2} - 115079983 T^{3} + 117316779389 T^{4} - 115079983 p^{3} T^{5} - 34710 p^{6} T^{6} + 13 p^{10} T^{7} + p^{12} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 - 313 T - 182548 T^{2} - 366476245 T^{3} + 433917127841 T^{4} - 366476245 p^{3} T^{5} - 182548 p^{6} T^{6} - 313 p^{9} T^{7} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 618 T + 1444919 T^{2} - 618 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $C_2^2:C_4$ | \( 1 + 129 T - 887727 T^{2} - 281187905 T^{3} + 774498117996 T^{4} - 281187905 p^{3} T^{5} - 887727 p^{6} T^{6} + 129 p^{9} T^{7} + p^{12} T^{8} \) |
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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
−9.341492491602280364006385676471, −8.905251466267889161111696215258, −8.765694251189707049730628489067, −8.680888160445391316985098786228, −8.619922732644784900094169310296, −7.52278906974417685364196105646, −7.40284273096934791159573837730, −7.26949282700427039308280485399, −7.14757337776636331592559815014, −6.71789094827573085665353338935, −6.20156805855498031212403487201, −5.86582336306415178007759262446, −5.72021075061519160272269438648, −5.41492568996749486670610834100, −5.00894664227608182555873429344, −4.78114795479683686875617868940, −4.73108826955017527231133941635, −3.26864786929740648160551938646, −3.22267302978225619941248886503, −3.16535805501212141740207372332, −3.15898287890370050312952666632, −1.91001395156876904032314582136, −1.71847873607972437168906726539, −1.11547517387112945469992172838, −0.989245781771151692467652935223,
0.989245781771151692467652935223, 1.11547517387112945469992172838, 1.71847873607972437168906726539, 1.91001395156876904032314582136, 3.15898287890370050312952666632, 3.16535805501212141740207372332, 3.22267302978225619941248886503, 3.26864786929740648160551938646, 4.73108826955017527231133941635, 4.78114795479683686875617868940, 5.00894664227608182555873429344, 5.41492568996749486670610834100, 5.72021075061519160272269438648, 5.86582336306415178007759262446, 6.20156805855498031212403487201, 6.71789094827573085665353338935, 7.14757337776636331592559815014, 7.26949282700427039308280485399, 7.40284273096934791159573837730, 7.52278906974417685364196105646, 8.619922732644784900094169310296, 8.680888160445391316985098786228, 8.765694251189707049730628489067, 8.905251466267889161111696215258, 9.341492491602280364006385676471
