| L(s) = 1 | + 16·2-s + 160·4-s + 1.28e3·8-s − 130·9-s + 628·11-s + 8.96e3·16-s − 2.08e3·18-s + 1.00e4·22-s + 2.62e3·23-s − 4.13e3·25-s + 1.77e4·29-s + 5.73e4·32-s − 2.08e4·36-s + 2.93e3·37-s + 9.83e3·43-s + 1.00e5·44-s + 4.19e4·46-s − 6.62e4·50-s + 1.55e3·53-s + 2.83e5·58-s + 3.44e5·64-s + 1.37e4·67-s − 5.66e4·71-s − 1.66e5·72-s + 4.69e4·74-s − 2.18e5·79-s − 6.65e4·81-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 5·4-s + 7.07·8-s − 0.534·9-s + 1.56·11-s + 35/4·16-s − 1.51·18-s + 4.42·22-s + 1.03·23-s − 1.32·25-s + 3.91·29-s + 9.89·32-s − 2.67·36-s + 0.352·37-s + 0.811·43-s + 7.82·44-s + 2.92·46-s − 3.74·50-s + 0.0758·53-s + 11.0·58-s + 21/2·64-s + 0.372·67-s − 1.33·71-s − 3.78·72-s + 0.995·74-s − 3.93·79-s − 1.12·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92236816 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92236816 ^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(48.27992916\) |
| \(L(\frac12)\) |
\(\approx\) |
\(48.27992916\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{2} T )^{4} \) |
| 7 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 130 T^{2} + 9274 p^{2} T^{4} + 130 p^{10} T^{6} + p^{20} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 + 4138 T^{2} + 15069186 T^{4} + 4138 p^{10} T^{6} + p^{20} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 314 T - 2962 T^{2} - 314 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 764578 T^{2} + 292159956066 T^{4} + 764578 p^{10} T^{6} + p^{20} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 691072 T^{2} - 1624510552254 T^{4} + 691072 p^{10} T^{6} + p^{20} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 404046 T^{2} + 1324192623914 T^{4} - 404046 p^{10} T^{6} + p^{20} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 1312 T + 7707614 T^{2} - 1312 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 8866 T + 60324074 T^{2} - 8866 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 92836468 T^{2} + 3695909786961558 T^{4} + 92836468 p^{10} T^{6} + p^{20} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 1466 T + 3677778 p T^{2} - 1466 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 276410032 T^{2} + 39064125351165858 T^{4} + 276410032 p^{10} T^{6} + p^{20} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 4918 T + 271736814 T^{2} - 4918 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 464693524 T^{2} + 120683872742016534 T^{4} + 464693524 p^{10} T^{6} + p^{20} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 776 T + 331555958 T^{2} - 776 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 2379976162 T^{2} + 2424748968577420938 T^{4} + 2379976162 p^{10} T^{6} + p^{20} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 283151910 T^{2} - 342708036903888286 T^{4} - 283151910 p^{10} T^{6} + p^{20} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 6852 T + 2598680678 T^{2} - 6852 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 28332 T + 170720206 T^{2} + 28332 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 2142879120 T^{2} + 7475057594391926306 T^{4} + 2142879120 p^{10} T^{6} + p^{20} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 109148 T + 8896028286 T^{2} + 109148 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 15554590978 T^{2} + 91511105301695799306 T^{4} + 15554590978 p^{10} T^{6} + p^{20} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 15004724512 T^{2} + \)\(11\!\cdots\!38\)\( T^{4} + 15004724512 p^{10} T^{6} + p^{20} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 26479159584 T^{2} + \)\(30\!\cdots\!94\)\( T^{4} + 26479159584 p^{10} T^{6} + p^{20} T^{8} \) |
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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
−9.395816945715691516452792316209, −8.795538125522190333896799543784, −8.558884578597917269499382537145, −8.403714632835815622402590570190, −8.136799792526876614766903681482, −7.37522415069425545467207416193, −7.11732524302885130130334612699, −7.05825486489827930430094875500, −6.88786647351916487689323530285, −6.05808312761816079770281385701, −5.98452206182231045438471435852, −5.91774915018348678732390577445, −5.79367732310639751712151807182, −4.77706749186716107888348740115, −4.67071730759147263260551424714, −4.49638234629936500941473503895, −4.40313472436031419715304435642, −3.52217481593148213759507034367, −3.40576797807359927572742357734, −3.09098188310153772574201062343, −2.64569535173045326377841719360, −2.16274258728705042962186279331, −1.70312578781183386513553561088, −1.02811778848200883786642880510, −0.73364883843848000188395354047,
0.73364883843848000188395354047, 1.02811778848200883786642880510, 1.70312578781183386513553561088, 2.16274258728705042962186279331, 2.64569535173045326377841719360, 3.09098188310153772574201062343, 3.40576797807359927572742357734, 3.52217481593148213759507034367, 4.40313472436031419715304435642, 4.49638234629936500941473503895, 4.67071730759147263260551424714, 4.77706749186716107888348740115, 5.79367732310639751712151807182, 5.91774915018348678732390577445, 5.98452206182231045438471435852, 6.05808312761816079770281385701, 6.88786647351916487689323530285, 7.05825486489827930430094875500, 7.11732524302885130130334612699, 7.37522415069425545467207416193, 8.136799792526876614766903681482, 8.403714632835815622402590570190, 8.558884578597917269499382537145, 8.795538125522190333896799543784, 9.395816945715691516452792316209
