| L(s) = 1 | + 3·2-s + 6·4-s − 3·5-s − 7-s + 10·8-s − 9·10-s + 5·11-s + 3·13-s − 3·14-s + 15·16-s − 5·17-s + 3·19-s − 18·20-s + 15·22-s + 23-s + 6·25-s + 9·26-s − 6·28-s + 10·29-s + 12·31-s + 21·32-s − 15·34-s + 3·35-s − 3·37-s + 9·38-s − 30·40-s + 6·41-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 3·4-s − 1.34·5-s − 0.377·7-s + 3.53·8-s − 2.84·10-s + 1.50·11-s + 0.832·13-s − 0.801·14-s + 15/4·16-s − 1.21·17-s + 0.688·19-s − 4.02·20-s + 3.19·22-s + 0.208·23-s + 6/5·25-s + 1.76·26-s − 1.13·28-s + 1.85·29-s + 2.15·31-s + 3.71·32-s − 2.57·34-s + 0.507·35-s − 0.493·37-s + 1.45·38-s − 4.74·40-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 5^{3} \cdot 37^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 5^{3} \cdot 37^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(21.99049976\) |
| \(L(\frac12)\) |
\(\approx\) |
\(21.99049976\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| 37 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 + T + 4 T^{2} + 13 T^{3} + 4 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 5 T + 34 T^{2} - 9 p T^{3} + 34 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 3 T + 35 T^{2} - 74 T^{3} + 35 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 5 T + 30 T^{2} + 9 p T^{3} + 30 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 3 T + 53 T^{2} - 110 T^{3} + 53 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - T + 9 T^{2} + 6 p T^{3} + 9 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 10 T + 104 T^{2} - 584 T^{3} + 104 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 12 T + 102 T^{2} - 636 T^{3} + 102 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 6 T + 44 T^{2} + 14 T^{3} + 44 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 16 T + 190 T^{2} - 1444 T^{3} + 190 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 16 T + 129 T^{2} + 768 T^{3} + 129 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 9 T + 128 T^{2} - 683 T^{3} + 128 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 14 T + 181 T^{2} - 1644 T^{3} + 181 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 2 T + 160 T^{2} + 182 T^{3} + 160 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 20 T + 273 T^{2} - 2744 T^{3} + 273 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 4 T + 157 T^{2} - 312 T^{3} + 157 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 17 T + 15 T^{2} + 1030 T^{3} + 15 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 34 T + 593 T^{2} - 6468 T^{3} + 593 p T^{4} - 34 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 19 T + 265 T^{2} + 2430 T^{3} + 265 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 9 T + 231 T^{2} - 1494 T^{3} + 231 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 12 T + 248 T^{2} - 1696 T^{3} + 248 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.74138961444963928999422567854, −7.06107639942601007484643409050, −6.89677858206849337570493242123, −6.88583131551328938441836809870, −6.56201587917051014614144666856, −6.32147799687527701090554089413, −6.30033301619035904332769654085, −5.84058599965658617352732801210, −5.58741157089753680495152057474, −5.23908378441880035264412197221, −4.83954014044841435874383885361, −4.64121054259790498806912908127, −4.62687534487359993510672443936, −4.11857067341460528949311028083, −4.06978361203335588335476122786, −3.66558351564406813982063191157, −3.46863694327578630058221255007, −3.14957058692443973291051858023, −3.12087774864937357790907931873, −2.29121100399843943369615663000, −2.28690693960173524393266366471, −2.11006660128168517913660186825, −1.07643003061251412209578338863, −0.907152712227820404555887963628, −0.808697552858476107938226420465,
0.808697552858476107938226420465, 0.907152712227820404555887963628, 1.07643003061251412209578338863, 2.11006660128168517913660186825, 2.28690693960173524393266366471, 2.29121100399843943369615663000, 3.12087774864937357790907931873, 3.14957058692443973291051858023, 3.46863694327578630058221255007, 3.66558351564406813982063191157, 4.06978361203335588335476122786, 4.11857067341460528949311028083, 4.62687534487359993510672443936, 4.64121054259790498806912908127, 4.83954014044841435874383885361, 5.23908378441880035264412197221, 5.58741157089753680495152057474, 5.84058599965658617352732801210, 6.30033301619035904332769654085, 6.32147799687527701090554089413, 6.56201587917051014614144666856, 6.88583131551328938441836809870, 6.89677858206849337570493242123, 7.06107639942601007484643409050, 7.74138961444963928999422567854
