| L(s) = 1 | + 6·2-s − 4-s − 96·8-s − 215·16-s − 66·17-s − 110·19-s − 6·23-s − 268·31-s + 378·32-s − 396·34-s − 660·38-s − 36·46-s − 1.11e3·47-s − 445·49-s − 1.82e3·53-s − 1.79e3·61-s − 1.60e3·62-s + 1.98e3·64-s + 66·68-s + 110·76-s + 682·79-s − 384·83-s + 6·92-s − 6.69e3·94-s − 2.67e3·98-s − 1.09e4·106-s − 6.21e3·107-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 1/8·4-s − 4.24·8-s − 3.35·16-s − 0.941·17-s − 1.32·19-s − 0.0543·23-s − 1.55·31-s + 2.08·32-s − 1.99·34-s − 2.81·38-s − 0.115·46-s − 3.46·47-s − 1.29·49-s − 4.72·53-s − 3.77·61-s − 3.29·62-s + 3.88·64-s + 0.117·68-s + 0.166·76-s + 0.971·79-s − 0.507·83-s + 0.00679·92-s − 7.34·94-s − 2.75·98-s − 10.0·106-s − 5.61·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_4$ | \( ( 1 - 3 T + 7 p T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 + 445 T^{2} + 102696 T^{4} + 445 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 - 1579 T^{2} + 4066944 T^{4} - 1579 p^{6} T^{6} + p^{12} T^{8} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 + 3352 T^{2} + 7169406 T^{4} + 3352 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 33 T + 8870 T^{2} + 33 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 55 T + 12600 T^{2} + 55 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 + 3 T + 11480 T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 + 90680 T^{2} + 3241919742 T^{4} + 90680 p^{6} T^{6} + p^{12} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 134 T + 34083 T^{2} + 134 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 + 175720 T^{2} + 12839678718 T^{4} + 175720 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 + 112316 T^{2} + 12490980774 T^{4} + 112316 p^{6} T^{6} + p^{12} T^{8} \) |
| 43 | $C_2^2 \wr C_2$ | \( 1 + 72292 T^{2} + 2591234646 T^{4} + 72292 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 558 T + 247934 T^{2} + 558 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 912 T + 489353 T^{2} + 912 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 163940 T^{2} + 70350239382 T^{4} + 163940 p^{6} T^{6} + p^{12} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 899 T + 655974 T^{2} + 899 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 + 73984 T^{2} + 60975608910 T^{4} + 73984 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 + 288896 T^{2} - 20733089826 T^{4} + 288896 p^{6} T^{6} + p^{12} T^{8} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 + 1007941 T^{2} + 508583134860 T^{4} + 1007941 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 341 T + 1004094 T^{2} - 341 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 + 192 T - 41035 T^{2} + 192 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 + 934844 T^{2} + 495118446054 T^{4} + 934844 p^{6} T^{6} + p^{12} T^{8} \) |
| 97 | $C_2^2 \wr C_2$ | \( 1 + 1272145 T^{2} + 1457332783008 T^{4} + 1272145 p^{6} T^{6} + p^{12} T^{8} \) |
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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
−7.65264184344506680072968552875, −7.47398946173202152194388187246, −6.69256049668903689888540579166, −6.68800812767243828431701406901, −6.66068258716907303196818780778, −6.38529116884500942379766196677, −6.02890416592763133843767125457, −5.76522416122077647442813495806, −5.75775042897308743253722896866, −5.11395637343069579937174399127, −4.99459814836587623724182988346, −4.90881311164136694351258315604, −4.87288025431690296824429800408, −4.42093711500680102501950890206, −4.14996802506420271241073259573, −4.08617185985768021584747718268, −3.88724818235611293604064921997, −3.21456635932494589346714796615, −3.19928813603066662686937778189, −3.18169044375320892871913009447, −2.79083822251682943417531686930, −1.98276231883436941238507234363, −1.96027919670111924256791156883, −1.43917331269640040388040704134, −1.26028489473052721435051228660, 0, 0, 0, 0,
1.26028489473052721435051228660, 1.43917331269640040388040704134, 1.96027919670111924256791156883, 1.98276231883436941238507234363, 2.79083822251682943417531686930, 3.18169044375320892871913009447, 3.19928813603066662686937778189, 3.21456635932494589346714796615, 3.88724818235611293604064921997, 4.08617185985768021584747718268, 4.14996802506420271241073259573, 4.42093711500680102501950890206, 4.87288025431690296824429800408, 4.90881311164136694351258315604, 4.99459814836587623724182988346, 5.11395637343069579937174399127, 5.75775042897308743253722896866, 5.76522416122077647442813495806, 6.02890416592763133843767125457, 6.38529116884500942379766196677, 6.66068258716907303196818780778, 6.68800812767243828431701406901, 6.69256049668903689888540579166, 7.47398946173202152194388187246, 7.65264184344506680072968552875
