| L(s) = 1 | − 3·7-s − 3·9-s + 3·11-s + 3·17-s + 3·19-s − 3·23-s − 3·27-s + 3·29-s + 6·31-s − 12·37-s − 6·41-s − 18·43-s − 15·47-s − 9·49-s − 6·53-s + 6·59-s + 27·61-s + 9·63-s − 12·67-s + 6·71-s + 15·73-s − 9·77-s + 12·83-s − 30·89-s − 9·97-s − 9·99-s + 15·103-s + ⋯ |
| L(s) = 1 | − 1.13·7-s − 9-s + 0.904·11-s + 0.727·17-s + 0.688·19-s − 0.625·23-s − 0.577·27-s + 0.557·29-s + 1.07·31-s − 1.97·37-s − 0.937·41-s − 2.74·43-s − 2.18·47-s − 9/7·49-s − 0.824·53-s + 0.781·59-s + 3.45·61-s + 1.13·63-s − 1.46·67-s + 0.712·71-s + 1.75·73-s − 1.02·77-s + 1.31·83-s − 3.17·89-s − 0.913·97-s − 0.904·99-s + 1.47·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9487590304\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9487590304\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + p T^{2} + p T^{3} + p^{2} T^{4} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 3 T + 18 T^{2} + 40 T^{3} + 18 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 3 T + 18 T^{2} - 78 T^{3} + 18 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 21 T^{2} - 29 T^{3} + 21 p T^{4} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 3 T + 36 T^{2} - 114 T^{3} + 36 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 3 T + 24 T^{2} - 162 T^{3} + 24 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 3 T + 81 T^{2} - 162 T^{3} + 81 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 6 T + 51 T^{2} - 123 T^{3} + 51 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{3} \) |
| 41 | $S_4\times C_2$ | \( 1 + 6 T + 63 T^{2} + 423 T^{3} + 63 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 18 T + 201 T^{2} + 1516 T^{3} + 201 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 15 T + 207 T^{2} + 1486 T^{3} + 207 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 6 T + 27 T^{2} + 100 T^{3} + 27 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 6 T + 141 T^{2} - 676 T^{3} + 141 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 27 T + 408 T^{2} - 3890 T^{3} + 408 p T^{4} - 27 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 12 T - 39 T^{2} - 1336 T^{3} - 39 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 6 T + 153 T^{2} - 649 T^{3} + 153 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 15 T + 177 T^{2} - 1602 T^{3} + 177 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 83 | $S_4\times C_2$ | \( 1 - 12 T + 189 T^{2} - 1928 T^{3} + 189 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 30 T + 531 T^{2} + 5948 T^{3} + 531 p T^{4} + 30 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 9 T + 222 T^{2} + 1608 T^{3} + 222 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.26874487083893868819458253128, −7.03550362378815540558693062602, −6.75554615938553402989547835012, −6.61867034052630599078262658942, −6.39973442969171879463051454901, −6.28969265132007406947399480446, −6.03336343033865885514520060135, −5.38469991865890374743231053395, −5.35846383151651284548569911974, −5.35764232093868238499225341117, −4.89399109171753279059364371614, −4.71053099322635979499286324920, −4.31636282896027431775560284745, −3.85335546859498625934956509892, −3.58358575374971499415875987371, −3.58011863683475866890809637968, −3.14721774688353596138073608584, −2.98551025108616736907057122416, −2.88427859794234836996290914228, −2.14792421892194592965075496680, −1.81052369145788542817098728257, −1.75932585083826764026719011128, −1.21450282551685192395197972281, −0.66896790272545048379324187365, −0.22160359030539828221124059868,
0.22160359030539828221124059868, 0.66896790272545048379324187365, 1.21450282551685192395197972281, 1.75932585083826764026719011128, 1.81052369145788542817098728257, 2.14792421892194592965075496680, 2.88427859794234836996290914228, 2.98551025108616736907057122416, 3.14721774688353596138073608584, 3.58011863683475866890809637968, 3.58358575374971499415875987371, 3.85335546859498625934956509892, 4.31636282896027431775560284745, 4.71053099322635979499286324920, 4.89399109171753279059364371614, 5.35764232093868238499225341117, 5.35846383151651284548569911974, 5.38469991865890374743231053395, 6.03336343033865885514520060135, 6.28969265132007406947399480446, 6.39973442969171879463051454901, 6.61867034052630599078262658942, 6.75554615938553402989547835012, 7.03550362378815540558693062602, 7.26874487083893868819458253128
