| L(s) = 1 | + 4·2-s + 3·3-s + 4·4-s + 9·5-s + 12·6-s − 19·7-s − 16·8-s − 21·9-s + 36·10-s + 24·11-s + 12·12-s − 61·13-s − 76·14-s + 27·15-s − 64·16-s + 6·17-s − 84·18-s + 266·19-s + 36·20-s − 57·21-s + 96·22-s − 69·23-s − 48·24-s + 34·25-s − 244·26-s − 72·27-s − 76·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 1/2·4-s + 0.804·5-s + 0.816·6-s − 1.02·7-s − 0.707·8-s − 7/9·9-s + 1.13·10-s + 0.657·11-s + 0.288·12-s − 1.30·13-s − 1.45·14-s + 0.464·15-s − 16-s + 0.0856·17-s − 1.09·18-s + 3.21·19-s + 0.402·20-s − 0.592·21-s + 0.930·22-s − 0.625·23-s − 0.408·24-s + 0.271·25-s − 1.84·26-s − 0.513·27-s − 0.512·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.221061581\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.221061581\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 - p T + 10 p T^{2} - p^{4} T^{3} + p^{6} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 9 T + 47 T^{2} + 1944 T^{3} - 24594 T^{4} + 1944 p^{3} T^{5} + 47 p^{6} T^{6} - 9 p^{9} T^{7} + p^{12} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 19 T - 179 T^{2} - 2774 T^{3} + 50128 T^{4} - 2774 p^{3} T^{5} - 179 p^{6} T^{6} + 19 p^{9} T^{7} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 24 T - 1285 T^{2} + 19224 T^{3} + 925104 T^{4} + 19224 p^{3} T^{5} - 1285 p^{6} T^{6} - 24 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 61 T - 1367 T^{2} + 42334 T^{3} + 12885898 T^{4} + 42334 p^{3} T^{5} - 1367 p^{6} T^{6} + 61 p^{9} T^{7} + p^{12} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 3 T + 9592 T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 7 p T + 12234 T^{2} - 7 p^{4} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 3 p T - 20527 T^{2} + 2862 p T^{3} + 433519968 T^{4} + 2862 p^{4} T^{5} - 20527 p^{6} T^{6} + 3 p^{10} T^{7} + p^{12} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 237 T + 4925 T^{2} + 584442 T^{3} + 661218474 T^{4} + 584442 p^{3} T^{5} + 4925 p^{6} T^{6} + 237 p^{9} T^{7} + p^{12} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 211 T - 24065 T^{2} + 1899844 T^{3} + 2490210604 T^{4} + 1899844 p^{3} T^{5} - 24065 p^{6} T^{6} + 211 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 262 T + 72162 T^{2} - 262 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 468 T + 27371 T^{2} + 25183548 T^{3} + 16885415064 T^{4} + 25183548 p^{3} T^{5} + 27371 p^{6} T^{6} + 468 p^{9} T^{7} + p^{12} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 2 p T - 17387 T^{2} + 268462 p T^{3} - 6295199732 T^{4} + 268462 p^{4} T^{5} - 17387 p^{6} T^{6} - 2 p^{10} T^{7} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 483 T + 20477 T^{2} + 2495178 T^{3} + 10288968168 T^{4} + 2495178 p^{3} T^{5} + 20477 p^{6} T^{6} + 483 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 150 T + 257074 T^{2} - 150 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 168 T - 388645 T^{2} + 1026648 T^{3} + 125802612624 T^{4} + 1026648 p^{3} T^{5} - 388645 p^{6} T^{6} + 168 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 1049 T + 424495 T^{2} - 232819256 T^{3} + 155558427094 T^{4} - 232819256 p^{3} T^{5} + 424495 p^{6} T^{6} - 1049 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 1166 T + 452161 T^{2} - 356643254 T^{3} + 324003162628 T^{4} - 356643254 p^{3} T^{5} + 452161 p^{6} T^{6} - 1166 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 312 T + 498238 T^{2} + 312 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 311 T + 698028 T^{2} + 311 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 349 T - 854801 T^{2} - 3307124 T^{3} + 650611367644 T^{4} - 3307124 p^{3} T^{5} - 854801 p^{6} T^{6} + 349 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 1221 T + 78743 T^{2} + 327867804 T^{3} + 814636885368 T^{4} + 327867804 p^{3} T^{5} + 78743 p^{6} T^{6} + 1221 p^{9} T^{7} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 492 T + 1092454 T^{2} + 492 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 128 T - 1698713 T^{2} + 14111872 T^{3} + 2093632480048 T^{4} + 14111872 p^{3} T^{5} - 1698713 p^{6} T^{6} - 128 p^{9} T^{7} + p^{12} T^{8} \) |
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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
−13.89662624150097310414775033587, −13.34376940576790711248211056619, −13.25250709963598213379394897253, −12.72052001372066200764273099936, −12.71408902175183919861051798423, −11.84939578941188341887547656442, −11.83011894895909056258898003511, −11.43944045384362872394399344443, −11.12470557929548630851373349131, −10.11058409929588419318488500924, −9.703925142424272870655477537562, −9.659404539468818438178579664217, −9.547834098055063183043822650011, −8.632252995053353520206374177106, −8.586624919062136920710889624436, −7.49280612897997457221569944911, −7.35951559821777938586083022979, −6.83613776844462763888020529733, −5.85119623212478453380424068911, −5.81611137140904375916041931556, −5.32425418152774817954800103619, −4.67515235124625793810476954388, −3.62768242447689383205795706978, −3.33354277687313836026402320475, −2.49126116116257301076434845504,
2.49126116116257301076434845504, 3.33354277687313836026402320475, 3.62768242447689383205795706978, 4.67515235124625793810476954388, 5.32425418152774817954800103619, 5.81611137140904375916041931556, 5.85119623212478453380424068911, 6.83613776844462763888020529733, 7.35951559821777938586083022979, 7.49280612897997457221569944911, 8.586624919062136920710889624436, 8.632252995053353520206374177106, 9.547834098055063183043822650011, 9.659404539468818438178579664217, 9.703925142424272870655477537562, 10.11058409929588419318488500924, 11.12470557929548630851373349131, 11.43944045384362872394399344443, 11.83011894895909056258898003511, 11.84939578941188341887547656442, 12.71408902175183919861051798423, 12.72052001372066200764273099936, 13.25250709963598213379394897253, 13.34376940576790711248211056619, 13.89662624150097310414775033587
