| L(s) = 1 | − 16·5-s + 16·13-s + 8·17-s + 108·25-s − 80·29-s + 16·37-s + 72·41-s + 180·49-s − 144·53-s − 176·61-s − 256·65-s + 40·73-s − 128·85-s − 136·89-s − 264·97-s − 208·101-s − 176·109-s − 328·113-s + 132·121-s − 432·125-s + 127-s + 131-s + 137-s + 139-s + 1.28e3·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 3.19·5-s + 1.23·13-s + 8/17·17-s + 4.31·25-s − 2.75·29-s + 0.432·37-s + 1.75·41-s + 3.67·49-s − 2.71·53-s − 2.88·61-s − 3.93·65-s + 0.547·73-s − 1.50·85-s − 1.52·89-s − 2.72·97-s − 2.05·101-s − 1.61·109-s − 2.90·113-s + 1.09·121-s − 3.45·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 8.82·145-s + 0.00671·149-s + 0.00662·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.02157643836\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02157643836\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( ( 1 + 8 T + 42 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 90 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 12 p T^{2} + 9062 T^{4} - 12 p^{5} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 8 T + 258 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 - 4 T + 198 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 1092 T^{2} + 534182 T^{4} - 1092 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 516 T^{2} + 998 p^{2} T^{4} - 516 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 40 T + 1482 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 1012 T^{2} + 112422 T^{4} - 1012 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 8 T + 1218 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 36 T + 3302 T^{2} - 36 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 900 T^{2} + 6425702 T^{4} - 900 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 2116 T^{2} + 7339782 T^{4} - 2116 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 72 T + 5738 T^{2} + 72 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 5762 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 88 T + 8994 T^{2} + 88 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 9156 T^{2} + 41992742 T^{4} - 9156 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 22417862 T^{4} - 132 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{4} \) |
| 79 | $D_4\times C_2$ | \( 1 - 6516 T^{2} + 24592550 T^{4} - 6516 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 13636 T^{2} + 114638502 T^{4} - 13636 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 68 T + 15462 T^{2} + 68 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 132 T + 21638 T^{2} + 132 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−6.14395161922198348454472357513, −6.12720467142908382261102874064, −5.92198900208118891268538536386, −5.54434820360929871715459929418, −5.30880967976045009580423914662, −5.20839565418313937306978203604, −4.90613893897824022175001862267, −4.53331839538180542088612902796, −4.39630367725870090070468889822, −4.14820758836851513516241260921, −4.08994213159793036001484222641, −3.76822802458099101720611924561, −3.60892003586653144188670186164, −3.52499712893403836850702346764, −3.51115450056841742785751836941, −2.82902890823686938854746640171, −2.68269918595148770658814110693, −2.40876382209212303335308704177, −2.40110886849802700726678536112, −1.50282588341295675273000643576, −1.39920039701782216762236992909, −1.24688137902644227225060261617, −0.961513046148331989969668791188, −0.19351709244857873202737003643, −0.05826042087447922638100226593,
0.05826042087447922638100226593, 0.19351709244857873202737003643, 0.961513046148331989969668791188, 1.24688137902644227225060261617, 1.39920039701782216762236992909, 1.50282588341295675273000643576, 2.40110886849802700726678536112, 2.40876382209212303335308704177, 2.68269918595148770658814110693, 2.82902890823686938854746640171, 3.51115450056841742785751836941, 3.52499712893403836850702346764, 3.60892003586653144188670186164, 3.76822802458099101720611924561, 4.08994213159793036001484222641, 4.14820758836851513516241260921, 4.39630367725870090070468889822, 4.53331839538180542088612902796, 4.90613893897824022175001862267, 5.20839565418313937306978203604, 5.30880967976045009580423914662, 5.54434820360929871715459929418, 5.92198900208118891268538536386, 6.12720467142908382261102874064, 6.14395161922198348454472357513
