| L(s) = 1 | + 4·2-s − 2·3-s + 10·4-s + 2·5-s − 8·6-s + 4·7-s + 20·8-s + 9-s + 8·10-s + 4·11-s − 20·12-s + 16·14-s − 4·15-s + 35·16-s + 2·17-s + 4·18-s + 2·19-s + 20·20-s − 8·21-s + 16·22-s + 4·23-s − 40·24-s + 25-s + 2·27-s + 40·28-s + 12·29-s − 16·30-s + ⋯ |
| L(s) = 1 | + 2.82·2-s − 1.15·3-s + 5·4-s + 0.894·5-s − 3.26·6-s + 1.51·7-s + 7.07·8-s + 1/3·9-s + 2.52·10-s + 1.20·11-s − 5.77·12-s + 4.27·14-s − 1.03·15-s + 35/4·16-s + 0.485·17-s + 0.942·18-s + 0.458·19-s + 4.47·20-s − 1.74·21-s + 3.41·22-s + 0.834·23-s − 8.16·24-s + 1/5·25-s + 0.384·27-s + 7.55·28-s + 2.22·29-s − 2.92·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(32.41338674\) |
| \(L(\frac12)\) |
\(\approx\) |
\(32.41338674\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 3 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 4 T + 3 T^{2} + 4 T^{3} + 8 T^{4} + 4 p T^{5} + 3 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 4 T - 5 T^{2} + 4 T^{3} + 144 T^{4} + 4 p T^{5} - 5 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 2 T + 14 T^{2} + 88 T^{3} - 257 T^{4} + 88 p T^{5} + 14 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 2 T - 30 T^{2} + 8 T^{3} + 719 T^{4} + 8 p T^{5} - 30 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 37 | $D_4\times C_2$ | \( 1 + 2 T - 66 T^{2} - 8 T^{3} + 3383 T^{4} - 8 p T^{5} - 66 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 2 T - 74 T^{2} + 8 T^{3} + 4239 T^{4} + 8 p T^{5} - 74 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 12 T + 42 T^{2} + 192 T^{3} + 2363 T^{4} + 192 p T^{5} + 42 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 6 T - 59 T^{2} + 66 T^{3} + 4308 T^{4} + 66 p T^{5} - 59 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 8 T - 65 T^{2} - 88 T^{3} + 9384 T^{4} - 88 p T^{5} - 65 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 6 T + 6 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 6 T - 102 T^{2} + 24 T^{3} + 12143 T^{4} + 24 p T^{5} - 102 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_4\times C_2$ | \( 1 + 4 T - 110 T^{2} - 64 T^{3} + 9699 T^{4} - 64 p T^{5} - 110 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 12 T + 42 T^{2} + 528 T^{3} - 5437 T^{4} + 528 p T^{5} + 42 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 12 T - 30 T^{2} - 192 T^{3} + 12659 T^{4} - 192 p T^{5} - 30 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 4 T - 109 T^{2} + 164 T^{3} + 7408 T^{4} + 164 p T^{5} - 109 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 4 T + 102 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 6 T + 183 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.12591824597114436194560558687, −6.67603705288710207954175819273, −6.61962140428452060413620881699, −6.38460600545476256265671725410, −6.28896702696444787260136976640, −5.96422958800387094707521819639, −5.60219747898408600335517062948, −5.55422950286914298306746318275, −5.40105213415700215547840144585, −5.15170737149064395230527577847, −4.80340319800111005895564784257, −4.79984101141725621638775107715, −4.63866726701474712718024119628, −4.16938149534862643675270086668, −3.84052534920316487701415817751, −3.84030815037765387422081616715, −3.54418667598113244431459708362, −3.03550976257335067274041279850, −2.87621108331005695076280069081, −2.43160366062712264929957063810, −2.34517647793929455924761992418, −1.77376832116667327366001240998, −1.54906714631110005559519872812, −1.16267744831252643065786330049, −0.907013957635431220924740999828,
0.907013957635431220924740999828, 1.16267744831252643065786330049, 1.54906714631110005559519872812, 1.77376832116667327366001240998, 2.34517647793929455924761992418, 2.43160366062712264929957063810, 2.87621108331005695076280069081, 3.03550976257335067274041279850, 3.54418667598113244431459708362, 3.84030815037765387422081616715, 3.84052534920316487701415817751, 4.16938149534862643675270086668, 4.63866726701474712718024119628, 4.79984101141725621638775107715, 4.80340319800111005895564784257, 5.15170737149064395230527577847, 5.40105213415700215547840144585, 5.55422950286914298306746318275, 5.60219747898408600335517062948, 5.96422958800387094707521819639, 6.28896702696444787260136976640, 6.38460600545476256265671725410, 6.61962140428452060413620881699, 6.67603705288710207954175819273, 7.12591824597114436194560558687
