| L(s) = 1 | + 2·2-s + 7·3-s − 4·4-s + 14·5-s + 14·6-s + 16·7-s − 37·8-s − 46·9-s + 28·10-s + 8·11-s − 28·12-s + 111·13-s + 32·14-s + 98·15-s − 70·16-s + 98·17-s − 92·18-s + 96·19-s − 56·20-s + 112·21-s + 16·22-s − 92·23-s − 259·24-s − 60·25-s + 222·26-s − 551·27-s − 64·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.34·3-s − 1/2·4-s + 1.25·5-s + 0.952·6-s + 0.863·7-s − 1.63·8-s − 1.70·9-s + 0.885·10-s + 0.219·11-s − 0.673·12-s + 2.36·13-s + 0.610·14-s + 1.68·15-s − 1.09·16-s + 1.39·17-s − 1.20·18-s + 1.15·19-s − 0.626·20-s + 1.16·21-s + 0.155·22-s − 0.834·23-s − 2.20·24-s − 0.479·25-s + 1.67·26-s − 3.92·27-s − 0.431·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 279841 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 279841 ^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.782191746\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.782191746\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 23 | $C_1$ | \( ( 1 + p T )^{4} \) |
| good | 2 | $C_2 \wr S_4$ | \( 1 - p T + p^{3} T^{2} + 13 T^{3} + p T^{4} + 13 p^{3} T^{5} + p^{9} T^{6} - p^{10} T^{7} + p^{12} T^{8} \) |
| 3 | $C_2 \wr S_4$ | \( 1 - 7 T + 95 T^{2} - 436 T^{3} + 3520 T^{4} - 436 p^{3} T^{5} + 95 p^{6} T^{6} - 7 p^{9} T^{7} + p^{12} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 - 14 T + 256 T^{2} - 418 T^{3} + 12846 T^{4} - 418 p^{3} T^{5} + 256 p^{6} T^{6} - 14 p^{9} T^{7} + p^{12} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 16 T + 204 T^{2} - 5384 T^{3} + 206630 T^{4} - 5384 p^{3} T^{5} + 204 p^{6} T^{6} - 16 p^{9} T^{7} + p^{12} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 8 T + 2836 T^{2} + 24208 T^{3} + 3924870 T^{4} + 24208 p^{3} T^{5} + 2836 p^{6} T^{6} - 8 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 111 T + 9317 T^{2} - 606318 T^{3} + 32607938 T^{4} - 606318 p^{3} T^{5} + 9317 p^{6} T^{6} - 111 p^{9} T^{7} + p^{12} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 98 T + 18644 T^{2} - 1340174 T^{3} + 134065526 T^{4} - 1340174 p^{3} T^{5} + 18644 p^{6} T^{6} - 98 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 96 T + 6228 T^{2} - 664352 T^{3} + 58340886 T^{4} - 664352 p^{3} T^{5} + 6228 p^{6} T^{6} - 96 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 21 T + 43029 T^{2} + 1233294 T^{3} + 1234620970 T^{4} + 1233294 p^{3} T^{5} + 43029 p^{6} T^{6} - 21 p^{9} T^{7} + p^{12} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 193 T + 118479 T^{2} + 15737568 T^{3} + 5226103696 T^{4} + 15737568 p^{3} T^{5} + 118479 p^{6} T^{6} + 193 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 170 T + 67600 T^{2} - 14886854 T^{3} + 4106178254 T^{4} - 14886854 p^{3} T^{5} + 67600 p^{6} T^{6} - 170 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 125 T + 262021 T^{2} + 25302094 T^{3} + 26646757314 T^{4} + 25302094 p^{3} T^{5} + 262021 p^{6} T^{6} + 125 p^{9} T^{7} + p^{12} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 2 T + 237596 T^{2} - 6518130 T^{3} + 25216368470 T^{4} - 6518130 p^{3} T^{5} + 237596 p^{6} T^{6} - 2 p^{9} T^{7} + p^{12} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 677 T + 441943 T^{2} + 176800520 T^{3} + 67040162064 T^{4} + 176800520 p^{3} T^{5} + 441943 p^{6} T^{6} + 677 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 230 T + 261896 T^{2} + 22166738 T^{3} + 41283664862 T^{4} + 22166738 p^{3} T^{5} + 261896 p^{6} T^{6} + 230 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 1140 T + 1222796 T^{2} + 746561364 T^{3} + 419058243382 T^{4} + 746561364 p^{3} T^{5} + 1222796 p^{6} T^{6} + 1140 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 754 T + 1043632 T^{2} - 513944326 T^{3} + 370106759150 T^{4} - 513944326 p^{3} T^{5} + 1043632 p^{6} T^{6} - 754 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 488 T + 942484 T^{2} - 309364928 T^{3} + 384185528102 T^{4} - 309364928 p^{3} T^{5} + 942484 p^{6} T^{6} - 488 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 401 T + 743943 T^{2} + 132494832 T^{3} + 270748693008 T^{4} + 132494832 p^{3} T^{5} + 743943 p^{6} T^{6} + 401 p^{9} T^{7} + p^{12} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 1509 T + 1981945 T^{2} - 1668507366 T^{3} + 1224656390878 T^{4} - 1668507366 p^{3} T^{5} + 1981945 p^{6} T^{6} - 1509 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 838 T + 1791072 T^{2} + 941948694 T^{3} + 1218053106718 T^{4} + 941948694 p^{3} T^{5} + 1791072 p^{6} T^{6} + 838 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 142 T + 1903548 T^{2} - 155539494 T^{3} + 1529982465222 T^{4} - 155539494 p^{3} T^{5} + 1903548 p^{6} T^{6} - 142 p^{9} T^{7} + p^{12} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 2342 T + 4281932 T^{2} - 4865072738 T^{3} + 4830100875446 T^{4} - 4865072738 p^{3} T^{5} + 4281932 p^{6} T^{6} - 2342 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 1062 T + 1869900 T^{2} - 2186621722 T^{3} + 1807325015910 T^{4} - 2186621722 p^{3} T^{5} + 1869900 p^{6} T^{6} - 1062 p^{9} T^{7} + p^{12} T^{8} \) |
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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
−13.41607913280787122031456034078, −12.89837041988241862465913436245, −12.47293221357823109433422292909, −12.00673288601360254869166702130, −11.48925967042603664859081290417, −11.39938497939304475551410015780, −11.26576041052549194571421720540, −10.55940034165964641102164597423, −9.975268380294332225708056230877, −9.508782662885469674936069842204, −9.392320644431521348886090010424, −8.855824625149566097471981464697, −8.738650172271792691317055559676, −8.263389515981525643308750721425, −7.894732367116790374871707639492, −7.71615961148401431843776613847, −6.32377992425189512401439173953, −6.21417340157474231380195744645, −5.80497233113977215700640049106, −5.43462989525735258810252461931, −4.97479405338250875487050898258, −3.60681355754534512502683995122, −3.57049243066173596144671526658, −2.92873018397997914043710060405, −1.85619229170449453679309191065,
1.85619229170449453679309191065, 2.92873018397997914043710060405, 3.57049243066173596144671526658, 3.60681355754534512502683995122, 4.97479405338250875487050898258, 5.43462989525735258810252461931, 5.80497233113977215700640049106, 6.21417340157474231380195744645, 6.32377992425189512401439173953, 7.71615961148401431843776613847, 7.894732367116790374871707639492, 8.263389515981525643308750721425, 8.738650172271792691317055559676, 8.855824625149566097471981464697, 9.392320644431521348886090010424, 9.508782662885469674936069842204, 9.975268380294332225708056230877, 10.55940034165964641102164597423, 11.26576041052549194571421720540, 11.39938497939304475551410015780, 11.48925967042603664859081290417, 12.00673288601360254869166702130, 12.47293221357823109433422292909, 12.89837041988241862465913436245, 13.41607913280787122031456034078
