| L(s) = 1 | − 2·2-s + 4·3-s + 2·4-s + 5·5-s − 8·6-s − 2·8-s + 9·9-s − 10·10-s − 4·11-s + 8·12-s − 3·13-s + 20·15-s + 4·17-s − 18·18-s + 7·19-s + 10·20-s + 8·22-s + 23-s − 8·24-s + 7·25-s + 6·26-s + 18·27-s − 7·29-s − 40·30-s − 3·31-s − 16·33-s − 8·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 2.30·3-s + 4-s + 2.23·5-s − 3.26·6-s − 0.707·8-s + 3·9-s − 3.16·10-s − 1.20·11-s + 2.30·12-s − 0.832·13-s + 5.16·15-s + 0.970·17-s − 4.24·18-s + 1.60·19-s + 2.23·20-s + 1.70·22-s + 0.208·23-s − 1.63·24-s + 7/5·25-s + 1.17·26-s + 3.46·27-s − 1.29·29-s − 7.30·30-s − 0.538·31-s − 2.78·33-s − 1.37·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.843265871\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.843265871\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + p T + p T^{2} + p T^{3} + p^{2} T^{4} + p^{3} T^{5} + p^{3} T^{6} \) |
| 3 | $S_4\times C_2$ | \( 1 - 4 T + 7 T^{2} - 10 T^{3} + 7 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - p T + 18 T^{2} - 43 T^{3} + 18 p T^{4} - p^{3} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 4 T + 3 p T^{2} + 86 T^{3} + 3 p^{2} T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 4 T + 49 T^{2} - 122 T^{3} + 49 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 7 T + 54 T^{2} - 203 T^{3} + 54 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - T + 28 T^{2} - 89 T^{3} + 28 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 7 T + 74 T^{2} + 403 T^{3} + 74 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 3 T + 52 T^{2} + 137 T^{3} + 52 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 10 T + 119 T^{2} + 658 T^{3} + 119 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 6 T + 7 T^{2} + 12 T^{3} + 7 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 9 T + 124 T^{2} - 673 T^{3} + 124 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 17 T + 230 T^{2} - 1745 T^{3} + 230 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 13 T + 198 T^{2} - 1369 T^{3} + 198 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 22 T + 321 T^{2} - 2848 T^{3} + 321 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 24 T + 343 T^{2} + 3152 T^{3} + 343 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 14 T + 165 T^{2} + 1228 T^{3} + 165 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 4 T + 169 T^{2} - 374 T^{3} + 169 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 5 T + 831 T^{3} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - T + 168 T^{2} - 257 T^{3} + 168 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 23 T + 376 T^{2} - 4021 T^{3} + 376 p T^{4} - 23 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 11 T + 212 T^{2} + 1979 T^{3} + 212 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 3 T + 220 T^{2} + 575 T^{3} + 220 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−9.529867127642993661590570600186, −9.182642743920545154488125772139, −8.889183321749360410690022288154, −8.785443725698670533903548305581, −8.431020605800283352557996123618, −8.000652481642021648717183770220, −7.67575127913523731317324166520, −7.44075378745038186326811013386, −7.29079128890212255960207671977, −7.11318411145537296109878140346, −6.62127784116218580160991895086, −5.89252159464686171875496880292, −5.80032554366248199096050883347, −5.43142123789935159051410651735, −5.38399324176441662561470680126, −4.73813545716517336757014977831, −4.24210765989283840943816093661, −3.73708896342617125646146836606, −3.18549583824662157349818568266, −3.13580267910941229711257118119, −2.35238640756927560759022767398, −2.28941462341848929539677406425, −2.14382562422290513776396535094, −1.47501913933754980833286348448, −0.821424707671999522381674432919,
0.821424707671999522381674432919, 1.47501913933754980833286348448, 2.14382562422290513776396535094, 2.28941462341848929539677406425, 2.35238640756927560759022767398, 3.13580267910941229711257118119, 3.18549583824662157349818568266, 3.73708896342617125646146836606, 4.24210765989283840943816093661, 4.73813545716517336757014977831, 5.38399324176441662561470680126, 5.43142123789935159051410651735, 5.80032554366248199096050883347, 5.89252159464686171875496880292, 6.62127784116218580160991895086, 7.11318411145537296109878140346, 7.29079128890212255960207671977, 7.44075378745038186326811013386, 7.67575127913523731317324166520, 8.000652481642021648717183770220, 8.431020605800283352557996123618, 8.785443725698670533903548305581, 8.889183321749360410690022288154, 9.182642743920545154488125772139, 9.529867127642993661590570600186
