| L(s) = 1 | − 3·5-s + 7·7-s − 3·11-s + 6·13-s + 5·17-s + 7·19-s + 3·23-s + 6·25-s + 29-s + 10·31-s − 21·35-s + 2·37-s + 10·41-s + 12·43-s − 47-s + 17·49-s − 10·53-s + 9·55-s − 10·59-s − 13·61-s − 18·65-s − 6·67-s − 10·71-s + 7·73-s − 21·77-s + 14·79-s − 16·83-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 2.64·7-s − 0.904·11-s + 1.66·13-s + 1.21·17-s + 1.60·19-s + 0.625·23-s + 6/5·25-s + 0.185·29-s + 1.79·31-s − 3.54·35-s + 0.328·37-s + 1.56·41-s + 1.82·43-s − 0.145·47-s + 17/7·49-s − 1.37·53-s + 1.21·55-s − 1.30·59-s − 1.66·61-s − 2.23·65-s − 0.733·67-s − 1.18·71-s + 0.819·73-s − 2.39·77-s + 1.57·79-s − 1.75·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{6} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{6} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.29136153\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.29136153\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| 23 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 - p T + 32 T^{2} - 102 T^{3} + 32 p T^{4} - p^{3} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 3 T + 32 T^{2} + 64 T^{3} + 32 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 6 T + 47 T^{2} - 157 T^{3} + 47 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 5 T + 20 T^{2} - 22 T^{3} + 20 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 7 T + 46 T^{2} - 160 T^{3} + 46 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - T - 3 T^{2} + 90 T^{3} - 3 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 10 T + 107 T^{2} - 567 T^{3} + 107 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $D_{6}$ | \( 1 - 2 T - 17 T^{2} + 364 T^{3} - 17 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 10 T + 75 T^{2} - 339 T^{3} + 75 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 12 T + 113 T^{2} - 776 T^{3} + 113 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + T + 135 T^{2} + 90 T^{3} + 135 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 10 T + 107 T^{2} + 1052 T^{3} + 107 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 10 T + 53 T^{2} - 4 T^{3} + 53 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 13 T + 154 T^{2} + 1372 T^{3} + 154 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 6 T + 41 T^{2} - 380 T^{3} + 41 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 10 T + 113 T^{2} + 545 T^{3} + 113 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 7 T + 185 T^{2} - 786 T^{3} + 185 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 14 T + 253 T^{2} - 1988 T^{3} + 253 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 16 T + 309 T^{2} + 2688 T^{3} + 309 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 20 T + 299 T^{2} - 3304 T^{3} + 299 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 5 T + 268 T^{2} - 914 T^{3} + 268 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.23454839419772069286180434097, −6.48875422682276677752665038972, −6.39779238876509865027056648751, −6.26351507911935154832854680165, −5.92305857251423681245221751532, −5.66912112607525873254510319463, −5.51370137570114927097568343727, −5.06489012155058622744354273476, −4.98473578067983757012324953544, −4.79253641677560298344890938299, −4.47184033676422710750795483654, −4.36454231660166562744923572010, −4.24745438364276635843989537102, −3.61417851928508586607543127149, −3.52700359740767183721304744288, −3.32264986205820500148066049159, −2.88445882692766116481773595049, −2.70100880977030274700456007458, −2.63118188237224067983570784240, −1.75996223251628944686473893423, −1.66800571469461125418401929365, −1.53059653701221511146746739057, −0.959429761585262024096309293536, −0.69214017199874804944040582976, −0.68180340617725493237626046191,
0.68180340617725493237626046191, 0.69214017199874804944040582976, 0.959429761585262024096309293536, 1.53059653701221511146746739057, 1.66800571469461125418401929365, 1.75996223251628944686473893423, 2.63118188237224067983570784240, 2.70100880977030274700456007458, 2.88445882692766116481773595049, 3.32264986205820500148066049159, 3.52700359740767183721304744288, 3.61417851928508586607543127149, 4.24745438364276635843989537102, 4.36454231660166562744923572010, 4.47184033676422710750795483654, 4.79253641677560298344890938299, 4.98473578067983757012324953544, 5.06489012155058622744354273476, 5.51370137570114927097568343727, 5.66912112607525873254510319463, 5.92305857251423681245221751532, 6.26351507911935154832854680165, 6.39779238876509865027056648751, 6.48875422682276677752665038972, 7.23454839419772069286180434097
