| L(s) = 1 | − 3·2-s + 2·3-s + 4·4-s + 3·5-s − 6·6-s − 4·8-s − 3·9-s − 9·10-s − 3·11-s + 8·12-s − 2·13-s + 6·15-s + 3·16-s + 9·18-s − 6·19-s + 12·20-s + 9·22-s − 10·23-s − 8·24-s + 6·25-s + 6·26-s − 10·27-s − 10·29-s − 18·30-s + 10·31-s + 32-s − 6·33-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 1.15·3-s + 2·4-s + 1.34·5-s − 2.44·6-s − 1.41·8-s − 9-s − 2.84·10-s − 0.904·11-s + 2.30·12-s − 0.554·13-s + 1.54·15-s + 3/4·16-s + 2.12·18-s − 1.37·19-s + 2.68·20-s + 1.91·22-s − 2.08·23-s − 1.63·24-s + 6/5·25-s + 1.17·26-s − 1.92·27-s − 1.85·29-s − 3.28·30-s + 1.79·31-s + 0.176·32-s − 1.04·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{3} \cdot 7^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{3} \cdot 7^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 - T )^{3} \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + 3 T + 5 T^{2} + 7 T^{3} + 5 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 3 | $S_4\times C_2$ | \( 1 - 2 T + 7 T^{2} - 10 T^{3} + 7 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 2 T + 17 T^{2} + 54 T^{3} + 17 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 21 T^{2} - 2 T^{3} + 21 p T^{4} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 6 T + 53 T^{2} + 220 T^{3} + 53 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 10 T + 97 T^{2} + 480 T^{3} + 97 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 10 T + 99 T^{2} + 540 T^{3} + 99 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 10 T + 113 T^{2} - 594 T^{3} + 113 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 16 T + 163 T^{2} + 1084 T^{3} + 163 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 87 T^{2} + 54 T^{3} + 87 p T^{4} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 2 T + 37 T^{2} + 440 T^{3} + 37 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 20 T + 251 T^{2} - 2038 T^{3} + 251 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 12 T + 179 T^{2} + 1276 T^{3} + 179 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 14 T + 3 p T^{2} + 1578 T^{3} + 3 p^{2} T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 10 T + 123 T^{2} + 1282 T^{3} + 123 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 2 T + 89 T^{2} + 96 T^{3} + 89 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 24 T + 293 T^{2} + 2608 T^{3} + 293 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 4 T + 61 T^{2} + 394 T^{3} + 61 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 8 T + 125 T^{2} - 1020 T^{3} + 125 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 10 T + 133 T^{2} - 564 T^{3} + 133 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 20 T + 315 T^{2} + 3240 T^{3} + 315 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 99 T^{2} + 160 T^{3} + 99 p T^{4} + p^{3} T^{6} \) |
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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
−8.462687250028694410236866108121, −7.910123526948680226167336961119, −7.899177443398826138501594175849, −7.79277555701009338172367694489, −7.43798747778658805353960405418, −7.09460439426064152008659609533, −6.69863013264155684944038014514, −6.54918024141490205210006027191, −6.12522952216754708416188007346, −6.05291780102710828405609301276, −5.65065923944090257488337823358, −5.49562919373982315358650642793, −5.34554628954152084259100760190, −4.71981354829861326197160257554, −4.46856700557285369464874371256, −4.28089293769313546113140376207, −3.61506901020544943838692872074, −3.54967966118980992598981860954, −2.89952424658914189787766920623, −2.83277108370639728297126259682, −2.45499810388981625316045482539, −2.30137955700654906792083694003, −1.84495277599261576460629129936, −1.65678267503887185000499362907, −1.17161553440110797841553710269, 0, 0, 0,
1.17161553440110797841553710269, 1.65678267503887185000499362907, 1.84495277599261576460629129936, 2.30137955700654906792083694003, 2.45499810388981625316045482539, 2.83277108370639728297126259682, 2.89952424658914189787766920623, 3.54967966118980992598981860954, 3.61506901020544943838692872074, 4.28089293769313546113140376207, 4.46856700557285369464874371256, 4.71981354829861326197160257554, 5.34554628954152084259100760190, 5.49562919373982315358650642793, 5.65065923944090257488337823358, 6.05291780102710828405609301276, 6.12522952216754708416188007346, 6.54918024141490205210006027191, 6.69863013264155684944038014514, 7.09460439426064152008659609533, 7.43798747778658805353960405418, 7.79277555701009338172367694489, 7.899177443398826138501594175849, 7.910123526948680226167336961119, 8.462687250028694410236866108121
