| L(s) = 1 | + 3·3-s − 2·7-s + 6·9-s + 2·11-s − 3·13-s − 2·17-s + 3·19-s − 6·21-s − 4·23-s + 10·27-s + 29-s + 14·31-s + 6·33-s − 37-s − 9·39-s + 5·41-s − 6·43-s + 5·47-s − 12·49-s − 6·51-s + 15·53-s + 9·57-s + 8·59-s + 18·61-s − 12·63-s − 3·67-s − 12·69-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.755·7-s + 2·9-s + 0.603·11-s − 0.832·13-s − 0.485·17-s + 0.688·19-s − 1.30·21-s − 0.834·23-s + 1.92·27-s + 0.185·29-s + 2.51·31-s + 1.04·33-s − 0.164·37-s − 1.44·39-s + 0.780·41-s − 0.914·43-s + 0.729·47-s − 1.71·49-s − 0.840·51-s + 2.06·53-s + 1.19·57-s + 1.04·59-s + 2.30·61-s − 1.51·63-s − 0.366·67-s − 1.44·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(12.49664683\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.49664683\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{3} \) |
| 5 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 + 2 T + 16 T^{2} + 20 T^{3} + 16 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 2 T + 4 T^{2} + 36 T^{3} + 4 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 2 T + 44 T^{2} + 64 T^{3} + 44 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 3 T + 17 T^{2} - 14 T^{3} + 17 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 4 T + 49 T^{2} + 120 T^{3} + 49 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - T + 54 T^{2} - 57 T^{3} + 54 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 14 T + 128 T^{2} - 768 T^{3} + 128 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + T + 87 T^{2} + 94 T^{3} + 87 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 5 T + 107 T^{2} - 346 T^{3} + 107 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 6 T + 101 T^{2} + 508 T^{3} + 101 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 5 T + 130 T^{2} - 411 T^{3} + 130 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 15 T + 194 T^{2} - 1579 T^{3} + 194 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 8 T + 80 T^{2} - 274 T^{3} + 80 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 18 T + 200 T^{2} - 1532 T^{3} + 200 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 3 T + 14 T^{2} + 341 T^{3} + 14 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 23 T + 365 T^{2} - 3502 T^{3} + 365 p T^{4} - 23 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 8 T + 207 T^{2} - 1088 T^{3} + 207 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 7 T + 125 T^{2} - 822 T^{3} + 125 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 8 T + 208 T^{2} - 1158 T^{3} + 208 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 14 T + 235 T^{2} + 2364 T^{3} + 235 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 9 T^{2} - 272 T^{3} - 9 p T^{4} + p^{3} T^{6} \) |
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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
−6.95226843690636668021618399339, −6.64830092603914128731069642449, −6.61487731310417875338173927364, −6.41796797197110059155321250359, −6.21762979626372457683138029856, −5.69828887806746847553125071080, −5.47864846213836003309431401787, −5.32203197887047253534596418842, −4.93953817050152635471610312513, −4.81300769640275445846210166649, −4.34253535925728150234271408013, −4.16442458450561634636238104478, −4.15295630319403242015120352751, −3.60212356180885519765312607053, −3.43796396340699461878803761398, −3.43568263029269544148650132176, −2.76464827656892912960537622508, −2.67104844372336206001247091008, −2.63587982411024090549093492802, −2.03533605584921324542492607317, −1.97163636720051956428110036566, −1.69822645751751309983858918802, −0.980656175413720460102212646079, −0.67744902604118656560256861552, −0.61467378067749353879409291721,
0.61467378067749353879409291721, 0.67744902604118656560256861552, 0.980656175413720460102212646079, 1.69822645751751309983858918802, 1.97163636720051956428110036566, 2.03533605584921324542492607317, 2.63587982411024090549093492802, 2.67104844372336206001247091008, 2.76464827656892912960537622508, 3.43568263029269544148650132176, 3.43796396340699461878803761398, 3.60212356180885519765312607053, 4.15295630319403242015120352751, 4.16442458450561634636238104478, 4.34253535925728150234271408013, 4.81300769640275445846210166649, 4.93953817050152635471610312513, 5.32203197887047253534596418842, 5.47864846213836003309431401787, 5.69828887806746847553125071080, 6.21762979626372457683138029856, 6.41796797197110059155321250359, 6.61487731310417875338173927364, 6.64830092603914128731069642449, 6.95226843690636668021618399339
