| L(s) = 1 | − 4·3-s + 8·5-s + 4·7-s + 10·9-s − 4·11-s − 32·15-s − 12·19-s − 16·21-s + 12·23-s + 30·25-s − 20·27-s + 8·29-s + 16·31-s + 16·33-s + 32·35-s + 12·41-s − 12·43-s + 80·45-s + 8·49-s + 16·53-s − 32·55-s + 48·57-s + 4·59-s + 40·63-s − 12·67-s − 48·69-s − 12·71-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 3.57·5-s + 1.51·7-s + 10/3·9-s − 1.20·11-s − 8.26·15-s − 2.75·19-s − 3.49·21-s + 2.50·23-s + 6·25-s − 3.84·27-s + 1.48·29-s + 2.87·31-s + 2.78·33-s + 5.40·35-s + 1.87·41-s − 1.82·43-s + 11.9·45-s + 8/7·49-s + 2.19·53-s − 4.31·55-s + 6.35·57-s + 0.520·59-s + 5.03·63-s − 1.46·67-s − 5.77·69-s − 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.838548073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.838548073\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 5 | $C_2$$\times$$C_2^2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 8 T^{2} + p^{2} T^{4} ) \) |
| 7 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 13 | $D_4\times C_2$ | \( 1 + 2 T^{2} + 48 T^{3} + 2 T^{4} + 48 p T^{5} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 + 12 T + 86 T^{2} + 492 T^{3} + 2402 T^{4} + 492 p T^{5} + 86 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 300 T^{3} + 1246 T^{4} - 300 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 8 T + 34 T^{2} - 200 T^{3} + 1026 T^{4} - 200 p T^{5} + 34 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 37 | $D_4\times C_2$ | \( 1 + 2 T^{2} + 144 T^{3} + 2 T^{4} + 144 p T^{5} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 516 T^{3} + 3694 T^{4} - 516 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 12 T + 38 T^{2} - 564 T^{3} - 6334 T^{4} - 564 p T^{5} + 38 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 486 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 16 T + 82 T^{2} + 128 T^{3} - 3294 T^{4} + 128 p T^{5} + 82 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 4 T + 22 T^{2} + 668 T^{3} - 2718 T^{4} + 668 p T^{5} + 22 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 2 T^{2} + 240 T^{3} + 2 T^{4} + 240 p T^{5} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 12 T + 86 T^{2} + 876 T^{3} + 7010 T^{4} + 876 p T^{5} + 86 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 876 T^{3} + 10654 T^{4} + 876 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 4 T + 22 T^{2} - 1052 T^{3} - 4254 T^{4} - 1052 p T^{5} + 22 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 516 T^{3} + 1582 T^{4} + 516 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 20 T + 222 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−8.203069342361807591084865522207, −7.24664716461413191978270049561, −7.20692931252585277806268664354, −6.99554326728272584941698287118, −6.86430037894820605605160467022, −6.46418763581828249252346575322, −6.33400766297038625218451985880, −5.99308155019659103855985239163, −5.93136916998507686226927125115, −5.62771630771575882347366221740, −5.44189980625816421228905083372, −5.24278620063758471246921778400, −5.15833910216022358580923054751, −4.50401667217769326936797604825, −4.43206726291277806129101364830, −4.37089362495940289305454240739, −4.31237424025358055801111098044, −3.09158296978737068721328779044, −2.77179358545574926568626883656, −2.58634571052367184445962326673, −2.41037668758587240086292507061, −1.64619902065811103944708740546, −1.63296884630190106820472100682, −1.29248485740921071513219556040, −0.66284398736771289767017551862,
0.66284398736771289767017551862, 1.29248485740921071513219556040, 1.63296884630190106820472100682, 1.64619902065811103944708740546, 2.41037668758587240086292507061, 2.58634571052367184445962326673, 2.77179358545574926568626883656, 3.09158296978737068721328779044, 4.31237424025358055801111098044, 4.37089362495940289305454240739, 4.43206726291277806129101364830, 4.50401667217769326936797604825, 5.15833910216022358580923054751, 5.24278620063758471246921778400, 5.44189980625816421228905083372, 5.62771630771575882347366221740, 5.93136916998507686226927125115, 5.99308155019659103855985239163, 6.33400766297038625218451985880, 6.46418763581828249252346575322, 6.86430037894820605605160467022, 6.99554326728272584941698287118, 7.20692931252585277806268664354, 7.24664716461413191978270049561, 8.203069342361807591084865522207
