| L(s) = 1 | − 3·2-s + 6·4-s + 2·7-s − 10·8-s − 2·11-s − 2·13-s − 6·14-s + 15·16-s + 4·17-s − 3·19-s + 6·22-s − 14·23-s + 6·26-s + 12·28-s − 14·29-s − 4·31-s − 21·32-s − 12·34-s − 17·37-s + 9·38-s + 8·41-s + 2·43-s − 12·44-s + 42·46-s − 13·47-s + 4·49-s − 12·52-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 3·4-s + 0.755·7-s − 3.53·8-s − 0.603·11-s − 0.554·13-s − 1.60·14-s + 15/4·16-s + 0.970·17-s − 0.688·19-s + 1.27·22-s − 2.91·23-s + 1.17·26-s + 2.26·28-s − 2.59·29-s − 0.718·31-s − 3.71·32-s − 2.05·34-s − 2.79·37-s + 1.45·38-s + 1.24·41-s + 0.304·43-s − 1.80·44-s + 6.19·46-s − 1.89·47-s + 4/7·49-s − 1.66·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1025390506\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1025390506\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 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
−7.07982581687921454678975047221, −6.67131918531023650799606680141, −6.61903363701869428033691230439, −6.37004817788632636368053731772, −5.86434721722882601978443053771, −5.72810776860239521708122215781, −5.62402299051102079954756523583, −5.31111932536376094648547630258, −5.10229104502581608875967570252, −5.09002319492617793987502708246, −4.33982975164361804492220988575, −4.06680172867987293925956239617, −4.00202739644310170640619634508, −3.71802031671785248718146147970, −3.38425539607140498469565534189, −3.22402136845271470381309951285, −2.70042094622137294120103889590, −2.30525392428765716410928896871, −2.21501035721447891472946580252, −1.94696767510668984163891284021, −1.76972506590258354513283606732, −1.51309853172338444837242952043, −0.970641260981495929752070411776, −0.53093777693241158996295069653, −0.10201992686446687274177013857,
0.10201992686446687274177013857, 0.53093777693241158996295069653, 0.970641260981495929752070411776, 1.51309853172338444837242952043, 1.76972506590258354513283606732, 1.94696767510668984163891284021, 2.21501035721447891472946580252, 2.30525392428765716410928896871, 2.70042094622137294120103889590, 3.22402136845271470381309951285, 3.38425539607140498469565534189, 3.71802031671785248718146147970, 4.00202739644310170640619634508, 4.06680172867987293925956239617, 4.33982975164361804492220988575, 5.09002319492617793987502708246, 5.10229104502581608875967570252, 5.31111932536376094648547630258, 5.62402299051102079954756523583, 5.72810776860239521708122215781, 5.86434721722882601978443053771, 6.37004817788632636368053731772, 6.61903363701869428033691230439, 6.67131918531023650799606680141, 7.07982581687921454678975047221
