| L(s) = 1 | + 5·3-s − 2·7-s − 19·9-s + 31·11-s − 8·13-s + 89·17-s + 87·19-s − 10·21-s − 122·23-s − 134·27-s + 84·29-s + 294·31-s + 155·33-s + 94·37-s − 40·39-s − 345·41-s − 412·43-s + 824·47-s − 333·49-s + 445·51-s − 74·53-s + 435·57-s + 1.04e3·59-s + 482·61-s + 38·63-s + 735·67-s − 610·69-s + ⋯ |
| L(s) = 1 | + 0.962·3-s − 0.107·7-s − 0.703·9-s + 0.849·11-s − 0.170·13-s + 1.26·17-s + 1.05·19-s − 0.103·21-s − 1.10·23-s − 0.955·27-s + 0.537·29-s + 1.70·31-s + 0.817·33-s + 0.417·37-s − 0.164·39-s − 1.31·41-s − 1.46·43-s + 2.55·47-s − 0.970·49-s + 1.22·51-s − 0.191·53-s + 1.01·57-s + 2.29·59-s + 1.01·61-s + 0.0759·63-s + 1.34·67-s − 1.06·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.039538245\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.039538245\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $S_4\times C_2$ | \( 1 - 5 T + 44 T^{2} - 181 T^{3} + 44 p^{3} T^{4} - 5 p^{6} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 2 T + 337 T^{2} - 4364 T^{3} + 337 p^{3} T^{4} + 2 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 31 T + 3508 T^{2} - 74047 T^{3} + 3508 p^{3} T^{4} - 31 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 8 T + 591 T^{2} - 156848 T^{3} + 591 p^{3} T^{4} + 8 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 89 T + 8734 T^{2} - 383789 T^{3} + 8734 p^{3} T^{4} - 89 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 87 T + 17172 T^{2} - 1201391 T^{3} + 17172 p^{3} T^{4} - 87 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 122 T + 37857 T^{2} + 2899940 T^{3} + 37857 p^{3} T^{4} + 122 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 84 T + 42447 T^{2} - 3818824 T^{3} + 42447 p^{3} T^{4} - 84 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 294 T + 89193 T^{2} - 15888508 T^{3} + 89193 p^{3} T^{4} - 294 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 94 T + 25619 T^{2} + 974636 T^{3} + 25619 p^{3} T^{4} - 94 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 345 T + 173478 T^{2} + 33313165 T^{3} + 173478 p^{3} T^{4} + 345 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 412 T + 230201 T^{2} + 56312872 T^{3} + 230201 p^{3} T^{4} + 412 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 824 T + 454493 T^{2} - 159690000 T^{3} + 454493 p^{3} T^{4} - 824 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 74 T + 242291 T^{2} + 22110396 T^{3} + 242291 p^{3} T^{4} + 74 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 1040 T + 622137 T^{2} - 287028320 T^{3} + 622137 p^{3} T^{4} - 1040 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 482 T + 516203 T^{2} - 138176204 T^{3} + 516203 p^{3} T^{4} - 482 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 735 T + 580956 T^{2} - 201276543 T^{3} + 580956 p^{3} T^{4} - 735 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 2048 T + 2303733 T^{2} - 1686995456 T^{3} + 2303733 p^{3} T^{4} - 2048 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 15 T + 940326 T^{2} + 7175365 T^{3} + 940326 p^{3} T^{4} - 15 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 998 T + 874777 T^{2} - 622163644 T^{3} + 874777 p^{3} T^{4} - 998 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 1221 T + 733620 T^{2} + 234081389 T^{3} + 733620 p^{3} T^{4} + 1221 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 1189 T + 1665246 T^{2} - 1034445553 T^{3} + 1665246 p^{3} T^{4} - 1189 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 1010 T - 724081 T^{2} + 1968025540 T^{3} - 724081 p^{3} T^{4} - 1010 p^{6} T^{5} + p^{9} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.577495031005727222158176195349, −8.361520482538412468765814851083, −8.199535904116042337523809371243, −8.183510587535800072915451686732, −7.64925496537324558470418262640, −7.40670906625698466295321917661, −6.93810111890689030596897397881, −6.67819338415075222026732587285, −6.47507352278813629030105555694, −6.19080660961479428700203040001, −5.49711489643803989228316229564, −5.41851303654566772689349711853, −5.41486836552192831609554856561, −4.68640421491316272443366487233, −4.41228934065329279660740498706, −4.00748951433385241408407471578, −3.42823359661501071830964070784, −3.35988001285977201632444601288, −3.25840072873704597316084668434, −2.51808713109159165827412259897, −2.13771642009261616638479911556, −2.10729250665132853016280611070, −1.02955149746387221732325745141, −0.998885137406518882225448563297, −0.46142392425618739727319560274,
0.46142392425618739727319560274, 0.998885137406518882225448563297, 1.02955149746387221732325745141, 2.10729250665132853016280611070, 2.13771642009261616638479911556, 2.51808713109159165827412259897, 3.25840072873704597316084668434, 3.35988001285977201632444601288, 3.42823359661501071830964070784, 4.00748951433385241408407471578, 4.41228934065329279660740498706, 4.68640421491316272443366487233, 5.41486836552192831609554856561, 5.41851303654566772689349711853, 5.49711489643803989228316229564, 6.19080660961479428700203040001, 6.47507352278813629030105555694, 6.67819338415075222026732587285, 6.93810111890689030596897397881, 7.40670906625698466295321917661, 7.64925496537324558470418262640, 8.183510587535800072915451686732, 8.199535904116042337523809371243, 8.361520482538412468765814851083, 8.577495031005727222158176195349
