| L(s) = 1 | + 3·2-s − 3·3-s + 6·4-s − 9·6-s + 2·7-s + 10·8-s + 6·9-s − 18·12-s + 4·13-s + 6·14-s + 15·16-s + 6·17-s + 18·18-s − 4·19-s − 6·21-s + 3·23-s − 30·24-s + 12·26-s − 10·27-s + 12·28-s + 14·29-s − 12·31-s + 21·32-s + 18·34-s + 36·36-s + 22·37-s − 12·38-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 1.73·3-s + 3·4-s − 3.67·6-s + 0.755·7-s + 3.53·8-s + 2·9-s − 5.19·12-s + 1.10·13-s + 1.60·14-s + 15/4·16-s + 1.45·17-s + 4.24·18-s − 0.917·19-s − 1.30·21-s + 0.625·23-s − 6.12·24-s + 2.35·26-s − 1.92·27-s + 2.26·28-s + 2.59·29-s − 2.15·31-s + 3.71·32-s + 3.08·34-s + 6·36-s + 3.61·37-s − 1.94·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{6} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{6} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(17.47586747\) |
| \(L(\frac12)\) |
\(\approx\) |
\(17.47586747\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 5 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 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 | $S_4\times C_2$ | \( 1 + 17 T^{2} - 16 T^{3} + 17 p T^{4} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 4 T + 31 T^{2} - 88 T^{3} + 31 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 6 T + 47 T^{2} - 196 T^{3} + 47 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 4 T + 49 T^{2} + 136 T^{3} + 49 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $D_{6}$ | \( 1 - 14 T + 91 T^{2} - 468 T^{3} + 91 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{3} \) |
| 37 | $S_4\times C_2$ | \( 1 - 22 T + 251 T^{2} - 1860 T^{3} + 251 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 6 T + 71 T^{2} + 500 T^{3} + 71 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 10 T + 125 T^{2} - 852 T^{3} + 125 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 8 T + 77 T^{2} + 496 T^{3} + 77 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 10 T + 179 T^{2} - 1052 T^{3} + 179 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 2 T + 93 T^{2} - 132 T^{3} + 93 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 14 T + 115 T^{2} + 596 T^{3} + 115 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 22 T + 341 T^{2} - 3180 T^{3} + 341 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 6 T + 89 T^{2} - 1084 T^{3} + 89 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 8 T + 187 T^{2} - 1040 T^{3} + 187 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 4 T + 93 T^{2} + 40 T^{3} + 93 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 20 T + 361 T^{2} - 3480 T^{3} + 361 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 8 T + 235 T^{2} - 1296 T^{3} + 235 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 6 T + 183 T^{2} + 916 T^{3} + 183 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.70041061516606118861642504678, −7.14653540184221539362429971485, −6.84081204107262263307714576313, −6.79147877174990468956288952741, −6.30413198794200263674705013442, −6.18235319323705270727623727729, −6.15835869421272157305638378557, −5.78425592947165822186236168181, −5.52462971686566551768726885880, −5.36320290076653303414790512635, −4.87274500937264886587868262380, −4.82551895769432277962631371305, −4.72280742586250913612426821968, −4.29330649336499824513522834704, −4.01813165253729848860822729333, −3.86979001322054675143154798486, −3.38571774773598197383550373651, −3.12321174842808345853439281654, −3.03290182485077204654775561443, −2.25473465911943442873994133452, −2.04536548292419714428597429239, −1.93244435438224387752000912313, −1.02454690257055292412493138090, −1.02176486620142303189577611982, −0.74009342190394486475844744794,
0.74009342190394486475844744794, 1.02176486620142303189577611982, 1.02454690257055292412493138090, 1.93244435438224387752000912313, 2.04536548292419714428597429239, 2.25473465911943442873994133452, 3.03290182485077204654775561443, 3.12321174842808345853439281654, 3.38571774773598197383550373651, 3.86979001322054675143154798486, 4.01813165253729848860822729333, 4.29330649336499824513522834704, 4.72280742586250913612426821968, 4.82551895769432277962631371305, 4.87274500937264886587868262380, 5.36320290076653303414790512635, 5.52462971686566551768726885880, 5.78425592947165822186236168181, 6.15835869421272157305638378557, 6.18235319323705270727623727729, 6.30413198794200263674705013442, 6.79147877174990468956288952741, 6.84081204107262263307714576313, 7.14653540184221539362429971485, 7.70041061516606118861642504678
