| L(s) = 1 | − 3·4-s + 2·13-s + 6·16-s − 4·17-s + 8·23-s + 14·25-s + 10·29-s − 8·43-s + 23·49-s − 6·52-s + 42·53-s − 10·61-s − 10·64-s + 12·68-s − 20·79-s − 24·92-s − 42·100-s − 2·101-s − 36·103-s + 28·107-s + 34·113-s − 30·116-s + 27·121-s + ⋯ |
| L(s) = 1 | − 3/2·4-s + 0.554·13-s + 3/2·16-s − 0.970·17-s + 1.66·23-s + 14/5·25-s + 1.85·29-s − 1.21·43-s + 23/7·49-s − 0.832·52-s + 5.76·53-s − 1.28·61-s − 5/4·64-s + 1.45·68-s − 2.25·79-s − 2.50·92-s − 4.19·100-s − 0.199·101-s − 3.54·103-s + 2.70·107-s + 3.19·113-s − 2.78·116-s + 2.45·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{24} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{24} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.667846340\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.667846340\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{3} \) |
| 3 | \( 1 \) |
| 13 | \( 1 - 2 T - 5 T^{2} + 20 T^{3} - 5 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| good | 5 | \( 1 - 14 T^{2} + 103 T^{4} - 544 T^{6} + 103 p^{2} T^{8} - 14 p^{4} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 - 23 T^{2} + 278 T^{4} - 2315 T^{6} + 278 p^{2} T^{8} - 23 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - 27 T^{2} + 270 T^{4} - 1987 T^{6} + 270 p^{2} T^{8} - 27 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( ( 1 + 2 T + 15 T^{2} - 40 T^{3} + 15 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 19 | \( 1 - 39 T^{2} + 1506 T^{4} - 29099 T^{6} + 1506 p^{2} T^{8} - 39 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( ( 1 - 4 T + 63 T^{2} - 166 T^{3} + 63 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( ( 1 - 5 T + 84 T^{2} - 281 T^{3} + 84 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( 1 - 146 T^{2} + 9791 T^{4} - 385868 T^{6} + 9791 p^{2} T^{8} - 146 p^{4} T^{10} + p^{6} T^{12} \) |
| 37 | \( 1 - 50 T^{2} + 2039 T^{4} - 114716 T^{6} + 2039 p^{2} T^{8} - 50 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( 1 - 26 T^{2} - 269 T^{4} + 51152 T^{6} - 269 p^{2} T^{8} - 26 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( ( 1 + 4 T + 89 T^{2} + 328 T^{3} + 89 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 - 234 T^{2} + 24687 T^{4} - 1493116 T^{6} + 24687 p^{2} T^{8} - 234 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 - 21 T + 258 T^{2} - 2199 T^{3} + 258 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 - 71 T^{2} + 1090 T^{4} - 25195 T^{6} + 1090 p^{2} T^{8} - 71 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( ( 1 + 5 T + 50 T^{2} + 971 T^{3} + 50 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 290 T^{2} + 40199 T^{4} - 3363836 T^{6} + 40199 p^{2} T^{8} - 290 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( 1 - 183 T^{2} + 20454 T^{4} - 1564603 T^{6} + 20454 p^{2} T^{8} - 183 p^{4} T^{10} + p^{6} T^{12} \) |
| 73 | \( 1 - 314 T^{2} + 45311 T^{4} - 4041644 T^{6} + 45311 p^{2} T^{8} - 314 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 10 T + 259 T^{2} + 1574 T^{3} + 259 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 347 T^{2} + 59758 T^{4} - 74945 p T^{6} + 59758 p^{2} T^{8} - 347 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( 1 - 450 T^{2} + 89379 T^{4} - 10190176 T^{6} + 89379 p^{2} T^{8} - 450 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( 1 - 30 T^{2} + 1515 T^{4} - 1010072 T^{6} + 1515 p^{2} T^{8} - 30 p^{4} T^{10} + p^{6} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.70210582525132362060560689192, −4.50950678399499793678080848284, −4.46248696673765329921653096501, −4.33054590902162298289017014308, −4.28344126351414951062977998907, −3.95172875693317266487826941955, −3.86269014703719583310749679921, −3.68678503585254253606825842306, −3.63624133188217399093129620695, −3.37670315783178198278306512524, −3.05798918027308906074519995228, −3.02068249966833165906844365450, −2.86414316361530903606490707485, −2.68040340949692152963727491799, −2.60642412092175438794862391174, −2.46135682274385866355924884339, −2.04931742346244181312500724322, −1.87139612750134877282762157668, −1.86350534227273560392577651706, −1.26330219257195615299773248059, −1.13806768194415315883905982845, −0.910326858207423649644440217636, −0.904911282590162549054224553599, −0.64898299550748686400781988610, −0.29173619884641526771046082871,
0.29173619884641526771046082871, 0.64898299550748686400781988610, 0.904911282590162549054224553599, 0.910326858207423649644440217636, 1.13806768194415315883905982845, 1.26330219257195615299773248059, 1.86350534227273560392577651706, 1.87139612750134877282762157668, 2.04931742346244181312500724322, 2.46135682274385866355924884339, 2.60642412092175438794862391174, 2.68040340949692152963727491799, 2.86414316361530903606490707485, 3.02068249966833165906844365450, 3.05798918027308906074519995228, 3.37670315783178198278306512524, 3.63624133188217399093129620695, 3.68678503585254253606825842306, 3.86269014703719583310749679921, 3.95172875693317266487826941955, 4.28344126351414951062977998907, 4.33054590902162298289017014308, 4.46248696673765329921653096501, 4.50950678399499793678080848284, 4.70210582525132362060560689192
Plot not available for L-functions of degree greater than 10.
