| L(s) = 1 | + 2-s + 4-s − 2·7-s + 3·8-s − 2·14-s + 16-s − 12·19-s + 6·25-s − 2·28-s + 8·29-s − 32-s − 12·37-s − 12·38-s − 8·47-s + 6·49-s + 6·50-s − 16·53-s − 6·56-s + 8·58-s + 16·59-s + 64-s − 12·74-s − 12·76-s − 8·83-s − 8·94-s + 6·98-s + 6·100-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.755·7-s + 1.06·8-s − 0.534·14-s + 1/4·16-s − 2.75·19-s + 6/5·25-s − 0.377·28-s + 1.48·29-s − 0.176·32-s − 1.97·37-s − 1.94·38-s − 1.16·47-s + 6/7·49-s + 0.848·50-s − 2.19·53-s − 0.801·56-s + 1.05·58-s + 2.08·59-s + 1/8·64-s − 1.39·74-s − 1.37·76-s − 0.878·83-s − 0.825·94-s + 0.606·98-s + 3/5·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.150534770\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.150534770\) |
| \(L(\frac{3}{2})\) |
|
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 | $D_{4}$ | \( 1 - T - p T^{3} + p^{2} T^{4} \) |
| 3 | | \( 1 \) |
| 7 | $D_{4}$ | \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| good | 5 | $C_2^2 \wr C_2$ | \( 1 - 6 T^{2} + 42 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 - 34 T^{2} + 514 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 - 12 T^{2} + 102 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 - 22 T^{2} + 682 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 - 82 T^{2} + 2722 T^{4} - 82 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $D_{4}$ | \( ( 1 + 6 T + 66 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 - 102 T^{2} + 5130 T^{4} - 102 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2 \wr C_2$ | \( 1 - 24 T^{2} + 3774 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 61 | $C_2^2 \wr C_2$ | \( 1 - 132 T^{2} + 10710 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 - 144 T^{2} + 10830 T^{4} - 144 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 - 114 T^{2} + 8418 T^{4} - 114 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 - 236 T^{2} + 24310 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 - 288 T^{2} + 33150 T^{4} - 288 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 - 294 T^{2} + 36618 T^{4} - 294 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2 \wr C_2$ | \( 1 - 204 T^{2} + 28950 T^{4} - 204 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−8.716570320669932295560392336736, −8.561676929671817800086832661022, −8.199201998403981279907580734221, −8.112271176030091713646032641211, −7.910688366249232273735482437063, −7.14431026890047472265958664964, −6.96874588769461866320379606334, −6.96528112629200188798302820996, −6.73505616909807984622477097375, −6.57083924130414393763449692854, −5.92388708571680527569796532004, −5.78057953225196759515243928198, −5.77868922247834442174551932415, −5.08843144426301272963333119371, −4.69304626545211694618803372859, −4.54999947629504821640751486100, −4.46711452391305228634967994700, −4.10920113436614441052851173917, −3.39538406667254509395169331491, −3.20508308914145837452820012099, −3.17823182755974289963550278791, −2.30709495294607646192895702078, −2.00998914018690913983320959374, −1.79586169949253477839784317123, −0.66884033543823757728129316142,
0.66884033543823757728129316142, 1.79586169949253477839784317123, 2.00998914018690913983320959374, 2.30709495294607646192895702078, 3.17823182755974289963550278791, 3.20508308914145837452820012099, 3.39538406667254509395169331491, 4.10920113436614441052851173917, 4.46711452391305228634967994700, 4.54999947629504821640751486100, 4.69304626545211694618803372859, 5.08843144426301272963333119371, 5.77868922247834442174551932415, 5.78057953225196759515243928198, 5.92388708571680527569796532004, 6.57083924130414393763449692854, 6.73505616909807984622477097375, 6.96528112629200188798302820996, 6.96874588769461866320379606334, 7.14431026890047472265958664964, 7.910688366249232273735482437063, 8.112271176030091713646032641211, 8.199201998403981279907580734221, 8.561676929671817800086832661022, 8.716570320669932295560392336736
