| L(s) = 1 | + 2·2-s + 4-s − 4·5-s − 3·7-s − 2·8-s − 8·10-s − 13-s − 6·14-s − 4·16-s + 2·17-s + 3·19-s − 4·20-s − 3·23-s + 10·25-s − 2·26-s − 3·28-s + 9·29-s − 2·31-s − 2·32-s + 4·34-s + 12·35-s + 6·38-s + 8·40-s + 8·41-s − 5·43-s − 6·46-s − 28·47-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s − 1.78·5-s − 1.13·7-s − 0.707·8-s − 2.52·10-s − 0.277·13-s − 1.60·14-s − 16-s + 0.485·17-s + 0.688·19-s − 0.894·20-s − 0.625·23-s + 2·25-s − 0.392·26-s − 0.566·28-s + 1.67·29-s − 0.359·31-s − 0.353·32-s + 0.685·34-s + 2.02·35-s + 0.973·38-s + 1.26·40-s + 1.24·41-s − 0.762·43-s − 0.884·46-s − 4.08·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5689526074\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5689526074\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{4} \) |
| 13 | $C_2^2$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| good | 7 | $D_4\times C_2$ | \( 1 + 3 T - 3 T^{2} - 6 T^{3} + 32 T^{4} - 6 p T^{5} - 3 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 9 T + 38 T^{2} - 9 p T^{3} + p^{2} T^{4} )( 1 + 9 T + 38 T^{2} + 9 p T^{3} + p^{2} T^{4} ) \) |
| 17 | $D_4\times C_2$ | \( 1 - 2 T - 14 T^{2} + 32 T^{3} - 33 T^{4} + 32 p T^{5} - 14 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 3 T - 27 T^{2} + 6 T^{3} + 764 T^{4} + 6 p T^{5} - 27 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 3 T - T^{2} - 108 T^{3} - 636 T^{4} - 108 p T^{5} - p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 9 T + 7 T^{2} - 144 T^{3} + 2286 T^{4} - 144 p T^{5} + 7 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + T + 24 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 - 57 T^{2} + 1880 T^{4} - 57 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 8 T + 34 T^{2} + 416 T^{3} - 3405 T^{4} + 416 p T^{5} + 34 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 5 T - 63 T^{2} + 10 T^{3} + 4820 T^{4} + 10 p T^{5} - 63 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{4} \) |
| 53 | $D_{4}$ | \( ( 1 + 13 T + 144 T^{2} + 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 17 T + 103 T^{2} - 1156 T^{3} + 14064 T^{4} - 1156 p T^{5} + 103 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 12 T + 42 T^{2} - 384 T^{3} - 3733 T^{4} - 384 p T^{5} + 42 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 18 T + 118 T^{2} + 1152 T^{3} + 14391 T^{4} + 1152 p T^{5} + 118 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 6 T + 2 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 19 T + 244 T^{2} - 19 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 6 T + 158 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 17 T + 145 T^{2} - 578 T^{3} - 12906 T^{4} - 578 p T^{5} + 145 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 6 T - 150 T^{2} + 48 T^{3} + 21695 T^{4} + 48 p T^{5} - 150 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.86256810924856419638733953742, −6.71807241265256384147783675249, −6.66728272190647394286437219377, −6.24839984934933855310698617700, −6.11531348065321135798161260861, −5.91921458071902263260351949777, −5.59013702407048418358423884709, −5.31184017457850902091929999632, −5.10652678720842929781212616922, −4.83485854949387857155647139973, −4.67313871087248402472005420313, −4.55822160272687823846799248988, −4.08339876811711341995963889825, −4.06784591873606981362631175324, −3.81770032688827370200436619814, −3.38033638654084112327032047813, −3.17559784853414327900779061942, −3.15931775262935576494851121271, −2.90947819592377417424378958648, −2.66605665155458832888053097825, −2.15174743755001189161073990169, −1.48234768449722870925741437243, −1.47827722220413331948394138890, −0.68842120080556210484386696999, −0.17336768521322871030647396929,
0.17336768521322871030647396929, 0.68842120080556210484386696999, 1.47827722220413331948394138890, 1.48234768449722870925741437243, 2.15174743755001189161073990169, 2.66605665155458832888053097825, 2.90947819592377417424378958648, 3.15931775262935576494851121271, 3.17559784853414327900779061942, 3.38033638654084112327032047813, 3.81770032688827370200436619814, 4.06784591873606981362631175324, 4.08339876811711341995963889825, 4.55822160272687823846799248988, 4.67313871087248402472005420313, 4.83485854949387857155647139973, 5.10652678720842929781212616922, 5.31184017457850902091929999632, 5.59013702407048418358423884709, 5.91921458071902263260351949777, 6.11531348065321135798161260861, 6.24839984934933855310698617700, 6.66728272190647394286437219377, 6.71807241265256384147783675249, 6.86256810924856419638733953742
