| L(s) = 1 | + 3-s + 4·11-s + 8·13-s + 10·17-s + 2·19-s + 3·23-s + 2·27-s + 3·31-s + 4·33-s − 6·37-s + 8·39-s − 11·41-s + 11·43-s + 4·47-s + 10·51-s − 14·53-s + 2·57-s + 5·59-s − 17·61-s − 20·67-s + 3·69-s − 19·71-s + 12·73-s − 79-s + 2·81-s + 28·83-s + 14·89-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.20·11-s + 2.21·13-s + 2.42·17-s + 0.458·19-s + 0.625·23-s + 0.384·27-s + 0.538·31-s + 0.696·33-s − 0.986·37-s + 1.28·39-s − 1.71·41-s + 1.67·43-s + 0.583·47-s + 1.40·51-s − 1.92·53-s + 0.264·57-s + 0.650·59-s − 2.17·61-s − 2.44·67-s + 0.361·69-s − 2.25·71-s + 1.40·73-s − 0.112·79-s + 2/9·81-s + 3.07·83-s + 1.48·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.69077760\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.69077760\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $S_4\times C_2$ | \( 1 - T + T^{2} - p T^{3} + p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 4 T + 30 T^{2} - 79 T^{3} + 30 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 8 T + 4 p T^{2} - 205 T^{3} + 4 p^{2} T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 10 T + 60 T^{2} - 259 T^{3} + 60 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 2 T + 34 T^{2} - 97 T^{3} + 34 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 3 T + 33 T^{2} - 57 T^{3} + 33 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 42 T^{2} + 81 T^{3} + 42 p T^{4} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 3 T + 51 T^{2} - 223 T^{3} + 51 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 6 T + 48 T^{2} + 295 T^{3} + 48 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 11 T + 45 T^{2} + 29 T^{3} + 45 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 11 T + 143 T^{2} - 875 T^{3} + 143 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 4 T + 120 T^{2} - 367 T^{3} + 120 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 14 T + 144 T^{2} + 935 T^{3} + 144 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 5 T + 3 p T^{2} - 581 T^{3} + 3 p^{2} T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 17 T + 209 T^{2} + 2075 T^{3} + 209 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 20 T + 301 T^{2} + 2752 T^{3} + 301 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 19 T + 300 T^{2} + 2743 T^{3} + 300 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{3} \) |
| 79 | $S_4\times C_2$ | \( 1 + T + 119 T^{2} + 607 T^{3} + 119 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 28 T + 378 T^{2} - 3667 T^{3} + 378 p T^{4} - 28 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 14 T + 306 T^{2} - 2447 T^{3} + 306 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 15 T + 273 T^{2} - 2905 T^{3} + 273 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.55985072088557596265244621512, −7.08570468313913497481575990335, −6.81423802954371866562652339443, −6.47508091455711464060013050742, −6.32002051565248116204196020800, −6.11977268728517215518274969414, −5.95384141751978717352919729475, −5.66060725235996422045256272934, −5.31579836155044320038721986199, −5.18438286160024668125836091872, −4.66495493502239004669541881899, −4.63508937404286853771097617534, −4.21634239445265908818637266635, −3.95547733328271807399583906941, −3.57486851815063695312542703647, −3.43514434255523654184716555761, −3.11733670338027567849879592937, −3.02414583807924712768416431294, −2.94398330989454825226693936485, −1.96029960676183560770132618758, −1.90473429883880932496372304393, −1.55526645713708545481800592781, −1.23324685342691919993915079721, −0.793486738776314626722320287314, −0.66222765961882594085234278878,
0.66222765961882594085234278878, 0.793486738776314626722320287314, 1.23324685342691919993915079721, 1.55526645713708545481800592781, 1.90473429883880932496372304393, 1.96029960676183560770132618758, 2.94398330989454825226693936485, 3.02414583807924712768416431294, 3.11733670338027567849879592937, 3.43514434255523654184716555761, 3.57486851815063695312542703647, 3.95547733328271807399583906941, 4.21634239445265908818637266635, 4.63508937404286853771097617534, 4.66495493502239004669541881899, 5.18438286160024668125836091872, 5.31579836155044320038721986199, 5.66060725235996422045256272934, 5.95384141751978717352919729475, 6.11977268728517215518274969414, 6.32002051565248116204196020800, 6.47508091455711464060013050742, 6.81423802954371866562652339443, 7.08570468313913497481575990335, 7.55985072088557596265244621512
