| L(s) = 1 | + 4·3-s + 10·9-s − 8·11-s − 4·13-s + 4·17-s − 12·19-s + 4·23-s + 20·27-s + 8·29-s − 16·31-s − 32·33-s − 8·37-s − 16·39-s − 4·41-s + 4·43-s − 12·47-s − 12·49-s + 16·51-s − 48·57-s − 8·59-s + 12·61-s − 24·67-s + 16·69-s − 12·71-s − 24·73-s − 12·79-s + 35·81-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 10/3·9-s − 2.41·11-s − 1.10·13-s + 0.970·17-s − 2.75·19-s + 0.834·23-s + 3.84·27-s + 1.48·29-s − 2.87·31-s − 5.57·33-s − 1.31·37-s − 2.56·39-s − 0.624·41-s + 0.609·43-s − 1.75·47-s − 1.71·49-s + 2.24·51-s − 6.35·57-s − 1.04·59-s + 1.53·61-s − 2.93·67-s + 1.92·69-s − 1.42·71-s − 2.80·73-s − 1.35·79-s + 35/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 5 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 50 T^{2} + 240 T^{3} + 866 T^{4} + 240 p T^{5} + 50 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr C_2\wr C_2$ | \( 1 - 4 T + 24 T^{2} + 20 T^{3} + 14 T^{4} + 20 p T^{5} + 24 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr C_2\wr C_2$ | \( 1 + 12 T + 88 T^{2} + 460 T^{3} + 2174 T^{4} + 460 p T^{5} + 88 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr C_2\wr C_2$ | \( 1 - 4 T + 48 T^{2} - 52 T^{3} + 926 T^{4} - 52 p T^{5} + 48 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 100 T^{2} + 504 T^{3} + 4646 T^{4} + 504 p T^{5} + 100 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr C_2\wr C_2$ | \( 1 + 4 T + 150 T^{2} + 436 T^{3} + 8930 T^{4} + 436 p T^{5} + 150 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr C_2\wr C_2$ | \( 1 - 4 T + 104 T^{2} - 132 T^{3} + 4734 T^{4} - 132 p T^{5} + 104 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr C_2\wr C_2$ | \( 1 + 12 T + 158 T^{2} + 1180 T^{3} + 9410 T^{4} + 1180 p T^{5} + 158 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr C_2\wr C_2$ | \( 1 + 132 T^{2} - 160 T^{3} + 8870 T^{4} - 160 p T^{5} + 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 242 T^{2} + 1392 T^{3} + 21602 T^{4} + 1392 p T^{5} + 242 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr C_2\wr C_2$ | \( 1 - 12 T + 200 T^{2} - 1460 T^{3} + 15486 T^{4} - 1460 p T^{5} + 200 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr C_2\wr C_2$ | \( 1 + 24 T + 404 T^{2} + 4888 T^{3} + 45030 T^{4} + 4888 p T^{5} + 404 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr C_2\wr C_2$ | \( 1 + 12 T + 238 T^{2} + 1980 T^{3} + 24226 T^{4} + 1980 p T^{5} + 238 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr C_2\wr C_2$ | \( 1 + 24 T + 404 T^{2} + 4488 T^{3} + 43318 T^{4} + 4488 p T^{5} + 404 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr C_2\wr C_2$ | \( 1 + 12 T + 184 T^{2} + 1948 T^{3} + 22862 T^{4} + 1948 p T^{5} + 184 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr C_2\wr C_2$ | \( 1 + 12 T + 126 T^{2} + 540 T^{3} + 2754 T^{4} + 540 p T^{5} + 126 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr C_2\wr C_2$ | \( 1 + 4 T + 166 T^{2} - 220 T^{3} + 13890 T^{4} - 220 p T^{5} + 166 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 340 T^{2} + 1944 T^{3} + 47126 T^{4} + 1944 p T^{5} + 340 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−5.87493998444514504985168377333, −5.55511650876490026472047832885, −5.45848866272016351322494616666, −5.23796987539083065825447717412, −5.19509844778754064970109531447, −4.85163003984853616928655857133, −4.62869307386204083014390507174, −4.60346793497560423384956087771, −4.50966629018661867398739041747, −4.05726985802655190485671254093, −4.01451061242531986994747150452, −3.80850160709359404240212361315, −3.57461700295822538598409860490, −3.16689340016298290364426851294, −3.10205004050751466307052271770, −3.04659524702784884811691368653, −2.89243853343559977716115035950, −2.47301913613901996225835688300, −2.41349048781533100077599982912, −2.34294414123532789363412968147, −2.08187610881511696666211467340, −1.60853851117380814363808442063, −1.46401820076699683352339139584, −1.32162337114402594317905037787, −1.28687868872561569782711769978, 0, 0, 0, 0,
1.28687868872561569782711769978, 1.32162337114402594317905037787, 1.46401820076699683352339139584, 1.60853851117380814363808442063, 2.08187610881511696666211467340, 2.34294414123532789363412968147, 2.41349048781533100077599982912, 2.47301913613901996225835688300, 2.89243853343559977716115035950, 3.04659524702784884811691368653, 3.10205004050751466307052271770, 3.16689340016298290364426851294, 3.57461700295822538598409860490, 3.80850160709359404240212361315, 4.01451061242531986994747150452, 4.05726985802655190485671254093, 4.50966629018661867398739041747, 4.60346793497560423384956087771, 4.62869307386204083014390507174, 4.85163003984853616928655857133, 5.19509844778754064970109531447, 5.23796987539083065825447717412, 5.45848866272016351322494616666, 5.55511650876490026472047832885, 5.87493998444514504985168377333
