| L(s) = 1 | − 3-s + 3·5-s − 3·7-s − 4·9-s + 3·11-s + 8·13-s − 3·15-s + 6·17-s − 6·19-s + 3·21-s + 3·23-s − 2·25-s + 5·27-s + 4·29-s + 5·31-s − 3·33-s − 9·35-s + 9·37-s − 8·39-s + 12·41-s − 10·43-s − 12·45-s + 2·47-s + 6·49-s − 6·51-s + 10·53-s + 9·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s − 1.13·7-s − 4/3·9-s + 0.904·11-s + 2.21·13-s − 0.774·15-s + 1.45·17-s − 1.37·19-s + 0.654·21-s + 0.625·23-s − 2/5·25-s + 0.962·27-s + 0.742·29-s + 0.898·31-s − 0.522·33-s − 1.52·35-s + 1.47·37-s − 1.28·39-s + 1.87·41-s − 1.52·43-s − 1.78·45-s + 0.291·47-s + 6/7·49-s − 0.840·51-s + 1.37·53-s + 1.21·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 7^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 7^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.220117083\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.220117083\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) |
| 11 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + T + 5 T^{2} + 4 T^{3} + 5 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - 3 T + 11 T^{2} - 26 T^{3} + 11 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 8 T + 53 T^{2} - 204 T^{3} + 53 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 6 T + 29 T^{2} - 4 p T^{3} + 29 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 6 T + 41 T^{2} + 164 T^{3} + 41 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 3 T + 65 T^{2} - 130 T^{3} + 65 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 4 T + 75 T^{2} - 216 T^{3} + 75 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 5 T + 77 T^{2} - 228 T^{3} + 77 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 9 T + 131 T^{2} - 674 T^{3} + 131 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 12 T + 105 T^{2} - 692 T^{3} + 105 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 10 T + 93 T^{2} + 628 T^{3} + 93 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 2 T + 63 T^{2} - 36 T^{3} + 63 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 10 T + 175 T^{2} - 1044 T^{3} + 175 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 3 T + 77 T^{2} + 596 T^{3} + 77 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 2 T + 169 T^{2} - 212 T^{3} + 169 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 13 T + 197 T^{2} + 1398 T^{3} + 197 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 21 T + 353 T^{2} - 3278 T^{3} + 353 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 24 T + 353 T^{2} - 3428 T^{3} + 353 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 83 | $S_4\times C_2$ | \( 1 + 10 T + 217 T^{2} + 1308 T^{3} + 217 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 23 T + 387 T^{2} - 4266 T^{3} + 387 p T^{4} - 23 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 3 T + 287 T^{2} + 578 T^{3} + 287 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.45995101503470156621529722112, −6.64903930571365948224229358191, −6.59168309916145689655098603093, −6.49936563852000211667899968417, −6.28966655949944458899053901129, −6.23208134586424517864760314755, −5.92048054958162461414789086924, −5.57743783480595021962188716712, −5.43836526770661077756662444711, −5.41879485226456040757612067951, −4.86420348683754742383728154386, −4.62284778795509380947977511878, −4.16030688020210300404618737318, −3.85866313175302265132462307389, −3.83296588637086814495489525105, −3.52636416905965683262555032562, −3.07745812826845890424959988324, −2.82024996874373740912321681378, −2.70642099406748424081350274144, −2.20275338133834264207170198226, −1.91331983094999731196658278395, −1.58049609864733983718843623101, −0.914920817811771079429097775206, −0.879416593546644066501743546962, −0.51964376121547142971706527419,
0.51964376121547142971706527419, 0.879416593546644066501743546962, 0.914920817811771079429097775206, 1.58049609864733983718843623101, 1.91331983094999731196658278395, 2.20275338133834264207170198226, 2.70642099406748424081350274144, 2.82024996874373740912321681378, 3.07745812826845890424959988324, 3.52636416905965683262555032562, 3.83296588637086814495489525105, 3.85866313175302265132462307389, 4.16030688020210300404618737318, 4.62284778795509380947977511878, 4.86420348683754742383728154386, 5.41879485226456040757612067951, 5.43836526770661077756662444711, 5.57743783480595021962188716712, 5.92048054958162461414789086924, 6.23208134586424517864760314755, 6.28966655949944458899053901129, 6.49936563852000211667899968417, 6.59168309916145689655098603093, 6.64903930571365948224229358191, 7.45995101503470156621529722112
