| L(s) = 1 | + 4·3-s − 2·7-s + 10·9-s + 4·11-s + 4·13-s − 8·17-s − 18·19-s − 8·21-s − 8·25-s + 20·27-s − 10·29-s − 12·31-s + 16·33-s − 2·37-s + 16·39-s + 2·41-s − 28·43-s − 6·47-s − 8·49-s − 32·51-s − 4·53-s − 72·57-s − 16·59-s − 10·61-s − 20·63-s − 10·71-s − 6·73-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 0.755·7-s + 10/3·9-s + 1.20·11-s + 1.10·13-s − 1.94·17-s − 4.12·19-s − 1.74·21-s − 8/5·25-s + 3.84·27-s − 1.85·29-s − 2.15·31-s + 2.78·33-s − 0.328·37-s + 2.56·39-s + 0.312·41-s − 4.26·43-s − 0.875·47-s − 8/7·49-s − 4.48·51-s − 0.549·53-s − 9.53·57-s − 2.08·59-s − 1.28·61-s − 2.51·63-s − 1.18·71-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 11^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 11^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 11 | $C_1$ | \( ( 1 - T )^{4} \) |
| 13 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 5 | $C_2 \wr S_4$ | \( 1 + 8 T^{2} - 14 T^{3} + 26 T^{4} - 14 p T^{5} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 + 2 T + 12 T^{2} - 2 T^{3} + 54 T^{4} - 2 p T^{5} + 12 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 8 T + 42 T^{2} + 246 T^{3} + 1262 T^{4} + 246 p T^{5} + 42 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 18 T + 180 T^{2} + 1206 T^{3} + 6054 T^{4} + 1206 p T^{5} + 180 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 48 T^{2} - 148 T^{3} + 1022 T^{4} - 148 p T^{5} + 48 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 10 T + 86 T^{2} + 488 T^{3} + 110 p T^{4} + 488 p T^{5} + 86 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 12 T + 104 T^{2} + 658 T^{3} + 3558 T^{4} + 658 p T^{5} + 104 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 2 T + 108 T^{2} + 286 T^{3} + 5286 T^{4} + 286 p T^{5} + 108 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 2 T + 112 T^{2} - 106 T^{3} + 5734 T^{4} - 106 p T^{5} + 112 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 28 T + 374 T^{2} + 3342 T^{3} + 23802 T^{4} + 3342 p T^{5} + 374 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 6 T + 104 T^{2} + 982 T^{3} + 5486 T^{4} + 982 p T^{5} + 104 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 4 T + 132 T^{2} + 412 T^{3} + 8390 T^{4} + 412 p T^{5} + 132 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 16 T + 140 T^{2} + 128 T^{3} - 1898 T^{4} + 128 p T^{5} + 140 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 10 T + 164 T^{2} + 742 T^{3} + 9750 T^{4} + 742 p T^{5} + 164 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 180 T^{2} - 18 T^{3} + 16790 T^{4} - 18 p T^{5} + 180 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 10 T + 208 T^{2} + 1994 T^{3} + 19582 T^{4} + 1994 p T^{5} + 208 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 6 T + 184 T^{2} + 22 p T^{3} + 16118 T^{4} + 22 p^{2} T^{5} + 184 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 8 T + 66 T^{2} - 514 T^{3} + 1962 T^{4} - 514 p T^{5} + 66 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 8 T + 212 T^{2} - 1368 T^{3} + 24806 T^{4} - 1368 p T^{5} + 212 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 6 T + 348 T^{2} - 1544 T^{3} + 46058 T^{4} - 1544 p T^{5} + 348 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 10 T + 140 T^{2} + 802 T^{3} - 4490 T^{4} + 802 p T^{5} + 140 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.16674892603518692111932810955, −5.88783007870307444912047054212, −5.58514366414757080122352898665, −5.50612760099681969398250898374, −5.16541382322275349671743567275, −4.68430746432600184955344960742, −4.65064022043898133402361622609, −4.63721565248165518575451355766, −4.55672738539071443727940978206, −3.98113670771878157020141866996, −3.96078721519521088802134518356, −3.83929436225159118674327730523, −3.77092645242538016556288018673, −3.37327674759947881768956129413, −3.36184157551822329992133840696, −3.23427138365908541094591424137, −2.95689695481874564368345763472, −2.43824346715164929885908782318, −2.23495471595432884863672251807, −2.19902167978871740264068120387, −2.12321517995728517777674257235, −1.71586815524691770531477982276, −1.49762024985916399532804230257, −1.46551508758138099942551267072, −1.32224825706532631857483839484, 0, 0, 0, 0,
1.32224825706532631857483839484, 1.46551508758138099942551267072, 1.49762024985916399532804230257, 1.71586815524691770531477982276, 2.12321517995728517777674257235, 2.19902167978871740264068120387, 2.23495471595432884863672251807, 2.43824346715164929885908782318, 2.95689695481874564368345763472, 3.23427138365908541094591424137, 3.36184157551822329992133840696, 3.37327674759947881768956129413, 3.77092645242538016556288018673, 3.83929436225159118674327730523, 3.96078721519521088802134518356, 3.98113670771878157020141866996, 4.55672738539071443727940978206, 4.63721565248165518575451355766, 4.65064022043898133402361622609, 4.68430746432600184955344960742, 5.16541382322275349671743567275, 5.50612760099681969398250898374, 5.58514366414757080122352898665, 5.88783007870307444912047054212, 6.16674892603518692111932810955
