L(s) = 1 | + 3·3-s − 15·5-s + 7·7-s + 27·9-s + 66·11-s + 11·13-s − 45·15-s + 198·17-s + 154·19-s + 21·21-s + 33·23-s + 298·25-s + 216·27-s + 51·29-s + 43·31-s + 198·33-s − 105·35-s − 100·37-s + 33·39-s − 132·41-s + 88·43-s − 405·45-s + 399·47-s + 624·49-s + 594·51-s + 108·53-s − 990·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.34·5-s + 0.377·7-s + 9-s + 1.80·11-s + 0.234·13-s − 0.774·15-s + 2.82·17-s + 1.85·19-s + 0.218·21-s + 0.299·23-s + 2.38·25-s + 1.53·27-s + 0.326·29-s + 0.249·31-s + 1.04·33-s − 0.507·35-s − 0.444·37-s + 0.135·39-s − 0.502·41-s + 0.312·43-s − 1.34·45-s + 1.23·47-s + 1.81·49-s + 1.63·51-s + 0.279·53-s − 2.42·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(7.772972838\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.772972838\) |
\(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 - p T - 2 p^{2} T^{2} - p^{4} T^{3} + p^{6} T^{4} \) |
good | 5 | $D_4\times C_2$ | \( 1 + 3 p T - 73 T^{2} + 144 p T^{3} + 45054 T^{4} + 144 p^{4} T^{5} - 73 p^{6} T^{6} + 3 p^{10} T^{7} + p^{12} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - p T - 575 T^{2} + 62 p T^{3} + 254920 T^{4} + 62 p^{4} T^{5} - 575 p^{6} T^{6} - p^{10} T^{7} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 6 p T + 103 p T^{2} - 306 p^{2} T^{3} + 23292 p^{2} T^{4} - 306 p^{5} T^{5} + 103 p^{7} T^{6} - 6 p^{10} T^{7} + p^{12} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 11 T - 2447 T^{2} + 20086 T^{3} + 1501978 T^{4} + 20086 p^{3} T^{5} - 2447 p^{6} T^{6} - 11 p^{9} T^{7} + p^{12} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 99 T + 11608 T^{2} - 99 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 77 T + 9186 T^{2} - 77 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 33 T - 893 p T^{2} + 89298 T^{3} + 306484632 T^{4} + 89298 p^{3} T^{5} - 893 p^{7} T^{6} - 33 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 51 T - 46819 T^{2} - 32742 T^{3} + 1784077290 T^{4} - 32742 p^{3} T^{5} - 46819 p^{6} T^{6} - 51 p^{9} T^{7} + p^{12} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 43 T - 58121 T^{2} - 16684 T^{3} + 2653813660 T^{4} - 16684 p^{3} T^{5} - 58121 p^{6} T^{6} - 43 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 50 T + 77874 T^{2} + 50 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 132 T - 45541 T^{2} - 9883764 T^{3} - 1986392520 T^{4} - 9883764 p^{3} T^{5} - 45541 p^{6} T^{6} + 132 p^{9} T^{7} + p^{12} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 88 T - 152909 T^{2} - 144232 T^{3} + 18872321152 T^{4} - 144232 p^{3} T^{5} - 152909 p^{6} T^{6} - 88 p^{9} T^{7} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 399 T - 19927 T^{2} + 11378682 T^{3} + 4778899632 T^{4} + 11378682 p^{3} T^{5} - 19927 p^{6} T^{6} - 399 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 54 T + 81970 T^{2} - 54 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 798 T + 219437 T^{2} - 5273982 T^{3} + 1228510332 T^{4} - 5273982 p^{3} T^{5} + 219437 p^{6} T^{6} - 798 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 439 T - 218465 T^{2} - 18778664 T^{3} + 73809546934 T^{4} - 18778664 p^{3} T^{5} - 218465 p^{6} T^{6} + 439 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 988 T + 166519 T^{2} - 205601812 T^{3} + 271446260584 T^{4} - 205601812 p^{3} T^{5} + 166519 p^{6} T^{6} - 988 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 1368 T + 1012606 T^{2} + 1368 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 455 T + 342636 T^{2} + 455 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 803 T + 285247 T^{2} - 503092348 T^{3} - 431718608228 T^{4} - 503092348 p^{3} T^{5} + 285247 p^{6} T^{6} + 803 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 813 T - 553393 T^{2} - 57550644 T^{3} + 769801212072 T^{4} - 57550644 p^{3} T^{5} - 553393 p^{6} T^{6} - 813 p^{9} T^{7} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 396 T + 1406374 T^{2} + 396 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 736 T - 1333241 T^{2} + 36498976 T^{3} + 2188025435632 T^{4} + 36498976 p^{3} T^{5} - 1333241 p^{6} T^{6} + 736 p^{9} T^{7} + p^{12} T^{8} \) |
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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
−8.923162093942473358849331630514, −8.743506940865656776600405341177, −8.561098076030799853035921119144, −8.462663131850112792387516976186, −8.118633066437556125728452135547, −7.36862341862922707283330044090, −7.35809569688147261725119880141, −7.30422517613570629653226156830, −7.17970444190504417471234050677, −6.82827228124520467608841839103, −6.01479841689630126358823714020, −5.85418209844255220901607772139, −5.81197236249359183508658473634, −5.12929493732082154523549869164, −4.68830360008834837642129462107, −4.59752437321974415223991188958, −4.19939187632612903026259997505, −3.59767299729359983755555429523, −3.55988541963852249054955840851, −3.09247529528569500167709330418, −2.97233492615663602100662270633, −2.07563713017263238471666359204, −1.22679688630984839371070331299, −0.992350996243523622954056875129, −0.937859501127751646444891802848,
0.937859501127751646444891802848, 0.992350996243523622954056875129, 1.22679688630984839371070331299, 2.07563713017263238471666359204, 2.97233492615663602100662270633, 3.09247529528569500167709330418, 3.55988541963852249054955840851, 3.59767299729359983755555429523, 4.19939187632612903026259997505, 4.59752437321974415223991188958, 4.68830360008834837642129462107, 5.12929493732082154523549869164, 5.81197236249359183508658473634, 5.85418209844255220901607772139, 6.01479841689630126358823714020, 6.82827228124520467608841839103, 7.17970444190504417471234050677, 7.30422517613570629653226156830, 7.35809569688147261725119880141, 7.36862341862922707283330044090, 8.118633066437556125728452135547, 8.462663131850112792387516976186, 8.561098076030799853035921119144, 8.743506940865656776600405341177, 8.923162093942473358849331630514