| L(s) = 1 | + 3·3-s − 3·5-s + 2·7-s + 6·9-s − 12·11-s + 4·13-s − 9·15-s − 2·17-s + 6·21-s − 4·23-s + 6·25-s + 10·27-s − 4·29-s + 3·31-s − 36·33-s − 6·35-s + 8·37-s + 12·39-s − 6·41-s + 12·43-s − 18·45-s − 8·47-s − 49-s − 6·51-s − 14·53-s + 36·55-s − 6·59-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 1.34·5-s + 0.755·7-s + 2·9-s − 3.61·11-s + 1.10·13-s − 2.32·15-s − 0.485·17-s + 1.30·21-s − 0.834·23-s + 6/5·25-s + 1.92·27-s − 0.742·29-s + 0.538·31-s − 6.26·33-s − 1.01·35-s + 1.31·37-s + 1.92·39-s − 0.937·41-s + 1.82·43-s − 2.68·45-s − 1.16·47-s − 1/7·49-s − 0.840·51-s − 1.92·53-s + 4.85·55-s − 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| 31 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 7 | $D_{6}$ | \( 1 - 2 T + 5 T^{2} - 32 T^{3} + 5 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{3} \) |
| 13 | $S_4\times C_2$ | \( 1 - 4 T + 27 T^{2} - 68 T^{3} + 27 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 2 T + 31 T^{2} + 76 T^{3} + 31 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 25 T^{2} + 32 T^{3} + 25 p T^{4} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 4 T + 45 T^{2} + 200 T^{3} + 45 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 4 T + 7 T^{2} + 20 T^{3} + 7 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 8 T + 91 T^{2} - 12 p T^{3} + 91 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 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 | $S_4\times C_2$ | \( 1 - 12 T + 145 T^{2} - 1000 T^{3} + 145 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 8 T + 85 T^{2} + 320 T^{3} + 85 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 14 T + 203 T^{2} + 1508 T^{3} + 203 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 6 T + 181 T^{2} + 704 T^{3} + 181 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 10 T + 147 T^{2} + 1148 T^{3} + 147 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 2 T + 145 T^{2} + 72 T^{3} + 145 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 18 T + 313 T^{2} + 2728 T^{3} + 313 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 20 T + 311 T^{2} + 2884 T^{3} + 311 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 8 T + 229 T^{2} + 1152 T^{3} + 229 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 47 T^{2} + 528 T^{3} - 47 p T^{4} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 195 T^{2} + 108 T^{3} + 195 p T^{4} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 2 T + 63 T^{2} + 1500 T^{3} + 63 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.64443528448979500558580674244, −7.30602137080135343775157827583, −7.07061318953925015870015784370, −6.78200538938929193632370576872, −6.30664349947775552813581024052, −6.12847064487138139419393417448, −6.07039735823532393487709708643, −5.53938500675363871540160882434, −5.47305353341659828757680509727, −5.04484264248308796552675820929, −4.82015368107901471006642433012, −4.63260262640938270307316705461, −4.49335105976835679105231285893, −4.12100657484608022503649979262, −3.97850898111803384755269849516, −3.63729274219292674509365783370, −3.26277742325363537465237649499, −3.03384813130667751458551599402, −2.96426601413158439588741422475, −2.48021929927858288697001189053, −2.45214303140907700228944496737, −2.23649355189918964092417479911, −1.52253217877685994716329365680, −1.32060071930241406493065857394, −1.24719348345048523538565314755, 0, 0, 0,
1.24719348345048523538565314755, 1.32060071930241406493065857394, 1.52253217877685994716329365680, 2.23649355189918964092417479911, 2.45214303140907700228944496737, 2.48021929927858288697001189053, 2.96426601413158439588741422475, 3.03384813130667751458551599402, 3.26277742325363537465237649499, 3.63729274219292674509365783370, 3.97850898111803384755269849516, 4.12100657484608022503649979262, 4.49335105976835679105231285893, 4.63260262640938270307316705461, 4.82015368107901471006642433012, 5.04484264248308796552675820929, 5.47305353341659828757680509727, 5.53938500675363871540160882434, 6.07039735823532393487709708643, 6.12847064487138139419393417448, 6.30664349947775552813581024052, 6.78200538938929193632370576872, 7.07061318953925015870015784370, 7.30602137080135343775157827583, 7.64443528448979500558580674244
