| L(s) = 1 | + 6·3-s + 21·9-s − 2·13-s + 12·17-s + 8·23-s + 10·25-s + 56·27-s − 4·29-s − 12·39-s + 24·43-s + 18·49-s + 72·51-s + 12·53-s − 12·61-s + 48·69-s + 60·75-s − 8·79-s + 126·81-s − 24·87-s − 4·101-s + 8·103-s + 16·107-s + 28·113-s − 42·117-s + 46·121-s + 127-s + 144·129-s + ⋯ |
| L(s) = 1 | + 3.46·3-s + 7·9-s − 0.554·13-s + 2.91·17-s + 1.66·23-s + 2·25-s + 10.7·27-s − 0.742·29-s − 1.92·39-s + 3.65·43-s + 18/7·49-s + 10.0·51-s + 1.64·53-s − 1.53·61-s + 5.77·69-s + 6.92·75-s − 0.900·79-s + 14·81-s − 2.57·87-s − 0.398·101-s + 0.788·103-s + 1.54·107-s + 2.63·113-s − 3.88·117-s + 4.18·121-s + 0.0887·127-s + 12.6·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(92.78031724\) |
| \(L(\frac12)\) |
\(\approx\) |
\(92.78031724\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - T )^{6} \) |
| 13 | \( 1 + 2 T + 3 T^{2} + 44 T^{3} + 3 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| good | 5 | \( 1 - 2 p T^{2} + 71 T^{4} - 396 T^{6} + 71 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 - 18 T^{2} + 143 T^{4} - 860 T^{6} + 143 p^{2} T^{8} - 18 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - 46 T^{2} + 1031 T^{4} - 14148 T^{6} + 1031 p^{2} T^{8} - 46 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( ( 1 - 2 T + p T^{2} )^{6} \) |
| 19 | \( 1 - 90 T^{2} + 3671 T^{4} - 88172 T^{6} + 3671 p^{2} T^{8} - 90 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( ( 1 - 4 T + 37 T^{2} - 120 T^{3} + 37 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( ( 1 + 2 T + 51 T^{2} + 108 T^{3} + 51 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( 1 - 130 T^{2} + 8255 T^{4} - 318396 T^{6} + 8255 p^{2} T^{8} - 130 p^{4} T^{10} + p^{6} T^{12} \) |
| 37 | \( 1 - 62 T^{2} + 1207 T^{4} - 16772 T^{6} + 1207 p^{2} T^{8} - 62 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( 1 - 66 T^{2} + 3471 T^{4} - 153916 T^{6} + 3471 p^{2} T^{8} - 66 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( ( 1 - 12 T + 65 T^{2} - 200 T^{3} + 65 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 - 198 T^{2} + 18911 T^{4} - 1107380 T^{6} + 18911 p^{2} T^{8} - 198 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 - 6 T + 59 T^{2} - 868 T^{3} + 59 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 - 286 T^{2} + 36327 T^{4} - 2706148 T^{6} + 36327 p^{2} T^{8} - 286 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( ( 1 + 6 T + 83 T^{2} + 68 T^{3} + 83 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 186 T^{2} + 20855 T^{4} - 1607852 T^{6} + 20855 p^{2} T^{8} - 186 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( 1 - 294 T^{2} + 41807 T^{4} - 3677492 T^{6} + 41807 p^{2} T^{8} - 294 p^{4} T^{10} + p^{6} T^{12} \) |
| 73 | \( 1 - 358 T^{2} + 58111 T^{4} - 5442580 T^{6} + 58111 p^{2} T^{8} - 358 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 4 T + 93 T^{2} + 568 T^{3} + 93 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 142 T^{2} + 16151 T^{4} - 1857348 T^{6} + 16151 p^{2} T^{8} - 142 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( 1 - 258 T^{2} + 30383 T^{4} - 2720060 T^{6} + 30383 p^{2} T^{8} - 258 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( 1 - 246 T^{2} + 35855 T^{4} - 3951284 T^{6} + 35855 p^{2} T^{8} - 246 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−4.54845351998761729990946847551, −4.31106905806513883840906410046, −4.22588643455383943050449842111, −4.15841631404298467756505381283, −3.96828223118110715427223692489, −3.90320669195303117504487994225, −3.58810330037055471259621394442, −3.43048130921961927126193057590, −3.36115776311611781855105010544, −3.15058784392505704625097638299, −3.01450841032220825087409868559, −2.97992994824898945449456947619, −2.95640394671098401907765118097, −2.60061862831530055935738289703, −2.51934451169219714609919823854, −2.27890810016696806848336557699, −2.11921754865425606766690288060, −1.95178036801997504904506563801, −1.91428769299826708713109071041, −1.51309434816253562202936834229, −1.25110313048238510998183428576, −0.968963723357926901051500389662, −0.956931438572964116759927026293, −0.815546196876631672865236839037, −0.54709752118419103150928856074,
0.54709752118419103150928856074, 0.815546196876631672865236839037, 0.956931438572964116759927026293, 0.968963723357926901051500389662, 1.25110313048238510998183428576, 1.51309434816253562202936834229, 1.91428769299826708713109071041, 1.95178036801997504904506563801, 2.11921754865425606766690288060, 2.27890810016696806848336557699, 2.51934451169219714609919823854, 2.60061862831530055935738289703, 2.95640394671098401907765118097, 2.97992994824898945449456947619, 3.01450841032220825087409868559, 3.15058784392505704625097638299, 3.36115776311611781855105010544, 3.43048130921961927126193057590, 3.58810330037055471259621394442, 3.90320669195303117504487994225, 3.96828223118110715427223692489, 4.15841631404298467756505381283, 4.22588643455383943050449842111, 4.31106905806513883840906410046, 4.54845351998761729990946847551
Plot not available for L-functions of degree greater than 10.
