| 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 − 259·24-s − 60·25-s + 222·26-s − 551·27-s + 64·28-s + 21·29-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 − 2.20·24-s − 0.479·25-s + 1.67·26-s − 3.92·27-s + 0.431·28-s + 0.134·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(23^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 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
−8.043221151327483623943854429201, −7.73039969296542860986111481823, −7.70173878075887255001952283561, −6.96590812517798797361763826880, −6.81626622599835596318067534148, −6.62235249095721870064250348694, −6.52036635073941858795456443576, −6.01937909236397689597919845042, −5.84545299346593497352413717896, −5.83051214560024905598016171348, −5.51132941293934779368041939068, −5.22092004566653856494916960005, −4.76045958973105609203139305775, −4.44734524408175940803082472618, −4.11795871920583298214171497312, −4.10463795098505723549789656333, −3.58028370247365592701336751607, −3.45878775380885906415778403098, −3.35459944258663402769512684220, −2.93120668967462588779538997095, −2.84490151438764186003636472226, −2.47782139645876126633744366480, −1.95339051485587450227408974080, −1.70640780935950325610822048526, −1.12445291568357287430798183834, 0, 0, 0, 0,
1.12445291568357287430798183834, 1.70640780935950325610822048526, 1.95339051485587450227408974080, 2.47782139645876126633744366480, 2.84490151438764186003636472226, 2.93120668967462588779538997095, 3.35459944258663402769512684220, 3.45878775380885906415778403098, 3.58028370247365592701336751607, 4.10463795098505723549789656333, 4.11795871920583298214171497312, 4.44734524408175940803082472618, 4.76045958973105609203139305775, 5.22092004566653856494916960005, 5.51132941293934779368041939068, 5.83051214560024905598016171348, 5.84545299346593497352413717896, 6.01937909236397689597919845042, 6.52036635073941858795456443576, 6.62235249095721870064250348694, 6.81626622599835596318067534148, 6.96590812517798797361763826880, 7.70173878075887255001952283561, 7.73039969296542860986111481823, 8.043221151327483623943854429201
