| L(s) = 1 | − 3·3-s − 2·7-s + 6·9-s − 4·11-s + 6·13-s + 4·17-s − 6·19-s + 6·21-s + 3·23-s + 25-s − 10·27-s − 2·29-s + 4·31-s + 12·33-s + 14·37-s − 18·39-s − 2·41-s + 2·43-s + 16·47-s − 5·49-s − 12·51-s − 4·53-s + 18·57-s + 12·59-s + 22·61-s − 12·63-s − 2·67-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.755·7-s + 2·9-s − 1.20·11-s + 1.66·13-s + 0.970·17-s − 1.37·19-s + 1.30·21-s + 0.625·23-s + 1/5·25-s − 1.92·27-s − 0.371·29-s + 0.718·31-s + 2.08·33-s + 2.30·37-s − 2.88·39-s − 0.312·41-s + 0.304·43-s + 2.33·47-s − 5/7·49-s − 1.68·51-s − 0.549·53-s + 2.38·57-s + 1.56·59-s + 2.81·61-s − 1.51·63-s − 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.310972525\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.310972525\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 23 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 5 | $D_{6}$ | \( 1 - T^{2} + 16 T^{3} - p T^{4} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 2 T + 9 T^{2} + 20 T^{3} + 9 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 4 T + 17 T^{2} + 56 T^{3} + 17 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 17 | $S_4\times C_2$ | \( 1 - 4 T + 43 T^{2} - 120 T^{3} + 43 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 6 T + 53 T^{2} + 220 T^{3} + 53 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 2 T + 35 T^{2} + 76 T^{3} + 35 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 4 T + 45 T^{2} - 184 T^{3} + 45 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 14 T + 139 T^{2} - 884 T^{3} + 139 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 2 T + 39 T^{2} + 60 T^{3} + 39 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 2 T + 77 T^{2} + 12 T^{3} + 77 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 16 T + 173 T^{2} - 1376 T^{3} + 173 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 4 T + 15 T^{2} - 168 T^{3} + 15 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 12 T + 161 T^{2} - 1352 T^{3} + 161 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 22 T + 307 T^{2} - 2884 T^{3} + 307 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 2 T + 149 T^{2} + 84 T^{3} + 149 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 8 T + 85 T^{2} - 1392 T^{3} + 85 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 79 | $S_4\times C_2$ | \( 1 - 26 T + 449 T^{2} - 4644 T^{3} + 449 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 4 T + 73 T^{2} - 504 T^{3} + 73 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 8 T + 227 T^{2} + 1120 T^{3} + 227 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 18 T + 335 T^{2} - 3196 T^{3} + 335 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.718651993633261235742664101418, −8.321131069440052930097597639254, −8.153908958737683657538972671777, −8.120686891753311086327396512929, −7.46955464318567561692288637246, −7.35914471000216979926393135577, −7.00156532335081058768886089002, −6.62623392310406154252545215878, −6.26435672737367452582093461553, −6.25622953462202610315779926184, −6.05852553180156122204560620487, −5.47955888416803786877964722493, −5.39022262323583417642282371797, −5.16886697608116189334735363430, −4.58225807142246093843214900691, −4.54397226023140078323647354917, −3.79086309884720451163885921534, −3.67897640785466255285382750889, −3.65542701300115364457106411977, −2.68544767092925493409204352717, −2.52612082283772817275418862988, −2.15872807674074029931745944345, −1.29022610918885736433243322314, −0.885677204869502899404185766881, −0.53551138840216696430331899125,
0.53551138840216696430331899125, 0.885677204869502899404185766881, 1.29022610918885736433243322314, 2.15872807674074029931745944345, 2.52612082283772817275418862988, 2.68544767092925493409204352717, 3.65542701300115364457106411977, 3.67897640785466255285382750889, 3.79086309884720451163885921534, 4.54397226023140078323647354917, 4.58225807142246093843214900691, 5.16886697608116189334735363430, 5.39022262323583417642282371797, 5.47955888416803786877964722493, 6.05852553180156122204560620487, 6.25622953462202610315779926184, 6.26435672737367452582093461553, 6.62623392310406154252545215878, 7.00156532335081058768886089002, 7.35914471000216979926393135577, 7.46955464318567561692288637246, 8.120686891753311086327396512929, 8.153908958737683657538972671777, 8.321131069440052930097597639254, 8.718651993633261235742664101418
