| L(s) = 1 | + 3·2-s + 2·3-s + 6·4-s + 6·6-s + 2·7-s + 10·8-s + 4·9-s + 2·11-s + 12·12-s − 2·13-s + 6·14-s + 15·16-s − 4·17-s + 12·18-s − 3·19-s + 4·21-s + 6·22-s + 14·23-s + 20·24-s − 6·26-s + 27-s + 12·28-s + 14·29-s − 4·31-s + 21·32-s + 4·33-s − 12·34-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 1.15·3-s + 3·4-s + 2.44·6-s + 0.755·7-s + 3.53·8-s + 4/3·9-s + 0.603·11-s + 3.46·12-s − 0.554·13-s + 1.60·14-s + 15/4·16-s − 0.970·17-s + 2.82·18-s − 0.688·19-s + 0.872·21-s + 1.27·22-s + 2.91·23-s + 4.08·24-s − 1.17·26-s + 0.192·27-s + 2.26·28-s + 2.59·29-s − 0.718·31-s + 3.71·32-s + 0.696·33-s − 2.05·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(22.84444621\) |
| \(L(\frac12)\) |
\(\approx\) |
\(22.84444621\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - 2 T + 7 T^{3} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 2 T + 3 p T^{3} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 2 T + T^{2} - 20 T^{3} + p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 2 T + 16 T^{2} + 73 T^{3} + 16 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 4 T + 36 T^{2} + 143 T^{3} + 36 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 14 T + 126 T^{2} - 707 T^{3} + 126 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 14 T + 128 T^{2} - 837 T^{3} + 128 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 4 T + 57 T^{2} + 96 T^{3} + 57 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 17 T + 174 T^{2} + 1249 T^{3} + 174 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 8 T + 75 T^{2} + 592 T^{3} + 75 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 2 T + 89 T^{2} - 244 T^{3} + 89 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 13 T + 116 T^{2} - 697 T^{3} + 116 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 10 T + 170 T^{2} + 1057 T^{3} + 170 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 6 T + 148 T^{2} + 533 T^{3} + 148 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 22 T + 255 T^{2} - 2148 T^{3} + 255 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 70 T^{2} - 7 p T^{3} + 70 p T^{4} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 2 T + 197 T^{2} + 260 T^{3} + 197 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 12 T + 192 T^{2} + 1509 T^{3} + 192 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 24 T + 349 T^{2} - 3472 T^{3} + 349 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 12 T + 85 T^{2} - 448 T^{3} + 85 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{3} \) |
| 97 | $S_4\times C_2$ | \( 1 + 111 T^{2} + 648 T^{3} + 111 p T^{4} + p^{3} T^{6} \) |
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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
−8.712602781636527498136243445687, −8.626159814920393922514024077340, −8.496889024136592306280115373498, −8.075894010530024812148858185685, −7.65862334787671301641646781821, −7.28025840620011693446166694910, −7.11204516473340676961782901409, −6.78734593420353788327942476600, −6.73853777901954683676849324847, −6.53049179238004494768702899980, −5.89772105921229902687536822883, −5.44986835051929002683014338050, −5.33188401267888014535227763496, −4.89242873982978485255593790771, −4.76445659554234147987757244775, −4.31480640408567750484737837572, −4.23652763277160090553895259888, −3.65140799252629995017406128079, −3.49589122725442479360417953773, −3.05165975502807886761171049816, −2.59022385295220915040614234210, −2.51140339553885340497874043869, −1.79621579347191580862917911994, −1.63930587319736415565967290568, −0.987570335650326101693547283034,
0.987570335650326101693547283034, 1.63930587319736415565967290568, 1.79621579347191580862917911994, 2.51140339553885340497874043869, 2.59022385295220915040614234210, 3.05165975502807886761171049816, 3.49589122725442479360417953773, 3.65140799252629995017406128079, 4.23652763277160090553895259888, 4.31480640408567750484737837572, 4.76445659554234147987757244775, 4.89242873982978485255593790771, 5.33188401267888014535227763496, 5.44986835051929002683014338050, 5.89772105921229902687536822883, 6.53049179238004494768702899980, 6.73853777901954683676849324847, 6.78734593420353788327942476600, 7.11204516473340676961782901409, 7.28025840620011693446166694910, 7.65862334787671301641646781821, 8.075894010530024812148858185685, 8.496889024136592306280115373498, 8.626159814920393922514024077340, 8.712602781636527498136243445687
