| L(s) = 1 | + 3-s − 3·5-s − 2·7-s + 9-s + 7·11-s + 13-s − 3·15-s + 10·17-s − 13·19-s − 2·21-s − 3·23-s + 6·25-s − 27-s − 13·29-s + 8·31-s + 7·33-s + 6·35-s − 5·37-s + 39-s + 8·41-s + 24·43-s − 3·45-s − 2·47-s − 9·49-s + 10·51-s − 53-s − 21·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s − 0.755·7-s + 1/3·9-s + 2.11·11-s + 0.277·13-s − 0.774·15-s + 2.42·17-s − 2.98·19-s − 0.436·21-s − 0.625·23-s + 6/5·25-s − 0.192·27-s − 2.41·29-s + 1.43·31-s + 1.21·33-s + 1.01·35-s − 0.821·37-s + 0.160·39-s + 1.24·41-s + 3.65·43-s − 0.447·45-s − 0.291·47-s − 9/7·49-s + 1.40·51-s − 0.137·53-s − 2.83·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.720488421\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.720488421\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| 23 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - T + 2 T^{3} - p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 2 T + 13 T^{2} + 27 T^{3} + 13 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 7 T + 40 T^{2} - 140 T^{3} + 40 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - T + 16 T^{2} + 24 T^{3} + 16 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 10 T + 61 T^{2} - 257 T^{3} + 61 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 13 T + 90 T^{2} + 432 T^{3} + 90 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 13 T + 113 T^{2} + 678 T^{3} + 113 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 8 T + 105 T^{2} - 495 T^{3} + 105 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 5 T + 19 T^{2} - 126 T^{3} + 19 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 8 T + 121 T^{2} - 655 T^{3} + 121 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{3} \) |
| 47 | $S_4\times C_2$ | \( 1 + 2 T + 49 T^{2} - 212 T^{3} + 49 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + T + 59 T^{2} + 406 T^{3} + 59 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 17 T + 243 T^{2} + 2042 T^{3} + 243 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 13 T + 132 T^{2} + 866 T^{3} + 132 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 5 T + 109 T^{2} - 174 T^{3} + 109 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 22 T + 309 T^{2} - 2899 T^{3} + 309 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 12 T + 43 T^{2} + 368 T^{3} + 43 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 4 T + 93 T^{2} - 120 T^{3} + 93 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 15 T + 289 T^{2} + 2454 T^{3} + 289 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 8 T + 139 T^{2} + 1360 T^{3} + 139 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 3 T + 210 T^{2} + 632 T^{3} + 210 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−6.81958857123440871293661366130, −6.65347211365554599901387542844, −6.56611384038250505653043893596, −6.54071180711964683928202502563, −6.06436684263477582493301994571, −5.77513251299169465228876921778, −5.74625787438582483704658803682, −5.40092164812512944042898637357, −5.10848134340511699269751344980, −4.62132126163058058408218737972, −4.29633131407904759341016497101, −4.17071427905431246841868577944, −4.13622685959038549297121858750, −3.77048350142978692165784547517, −3.63097656764416510019899283663, −3.53673367842698188455855425497, −2.85005657271218686986235037289, −2.84938167841183354427928620375, −2.62619672697677190812160725695, −2.04598562968674211633358081153, −1.71884993646833810372214609288, −1.41722309508015648566065006397, −1.24340931689433925516283706346, −0.58087134435570794693660983215, −0.33867661168378627352240419280,
0.33867661168378627352240419280, 0.58087134435570794693660983215, 1.24340931689433925516283706346, 1.41722309508015648566065006397, 1.71884993646833810372214609288, 2.04598562968674211633358081153, 2.62619672697677190812160725695, 2.84938167841183354427928620375, 2.85005657271218686986235037289, 3.53673367842698188455855425497, 3.63097656764416510019899283663, 3.77048350142978692165784547517, 4.13622685959038549297121858750, 4.17071427905431246841868577944, 4.29633131407904759341016497101, 4.62132126163058058408218737972, 5.10848134340511699269751344980, 5.40092164812512944042898637357, 5.74625787438582483704658803682, 5.77513251299169465228876921778, 6.06436684263477582493301994571, 6.54071180711964683928202502563, 6.56611384038250505653043893596, 6.65347211365554599901387542844, 6.81958857123440871293661366130
