| L(s) = 1 | − 3·3-s − 3·5-s + 8-s + 6·9-s + 6·11-s − 9·13-s + 9·15-s − 6·17-s + 6·19-s − 3·23-s − 3·24-s − 3·25-s − 10·27-s + 6·29-s + 18·31-s − 18·33-s + 6·37-s + 27·39-s − 3·40-s + 6·41-s − 9·43-s − 18·45-s − 18·47-s + 18·51-s + 3·53-s − 18·55-s − 18·57-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.34·5-s + 0.353·8-s + 2·9-s + 1.80·11-s − 2.49·13-s + 2.32·15-s − 1.45·17-s + 1.37·19-s − 0.625·23-s − 0.612·24-s − 3/5·25-s − 1.92·27-s + 1.11·29-s + 3.23·31-s − 3.13·33-s + 0.986·37-s + 4.32·39-s − 0.474·40-s + 0.937·41-s − 1.37·43-s − 2.68·45-s − 2.62·47-s + 2.52·51-s + 0.412·53-s − 2.42·55-s − 2.38·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{6} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{6} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4989053583\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4989053583\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + T )^{3} \) |
| 7 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $D_{6}$ | \( 1 - T^{3} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 3 T + 12 T^{2} + 24 T^{3} + 12 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 6 T + 30 T^{2} - 112 T^{3} + 30 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 9 T + 54 T^{2} + 240 T^{3} + 54 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 6 T + 42 T^{2} + 154 T^{3} + 42 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 6 T + 54 T^{2} - 208 T^{3} + 54 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 6 T + 42 T^{2} - 86 T^{3} + 42 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 18 T + 6 p T^{2} - 1244 T^{3} + 6 p^{2} T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 6 T + 66 T^{2} - 494 T^{3} + 66 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 6 T + 30 T^{2} + 98 T^{3} + 30 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 9 T + 126 T^{2} + 650 T^{3} + 126 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 18 T + 201 T^{2} + 1500 T^{3} + 201 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 3 T + 156 T^{2} - 312 T^{3} + 156 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 3 T + 120 T^{2} + 342 T^{3} + 120 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 3 T + 156 T^{2} - 276 T^{3} + 156 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 21 T + 228 T^{2} + 1906 T^{3} + 228 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 9 T + 90 T^{2} - 854 T^{3} + 90 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 12 T + 192 T^{2} + 1762 T^{3} + 192 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 12 T + 228 T^{2} - 1576 T^{3} + 228 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 12 T + 276 T^{2} + 1956 T^{3} + 276 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 15 T + 330 T^{2} + 2720 T^{3} + 330 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 12 T + 264 T^{2} - 1846 T^{3} + 264 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.49696345504777717971592940945, −7.39663048840699601049389016299, −7.20663605520152693178792292834, −6.71455492544639511164609880072, −6.56566879773725328895352107335, −6.41138665147779275924755758272, −6.28862521698114964792106375290, −6.00735878307250279328760243835, −5.42033306434568974412527810687, −5.36270397000094329740400296419, −4.93496312411866146404560896215, −4.66245048605440427534088636445, −4.55477257525123512930064450708, −4.34153736690433087421148271903, −4.08227951452974919316051352725, −4.01559060560702687459658896699, −3.25473597799137279827818728926, −3.08399025486773192999829271992, −2.83179633898660359022021104389, −2.31945273943292824247292950738, −1.95685428237061879761503200943, −1.39537306277300571392664907869, −1.26913325991095883801150110616, −0.59975001852631606316816543160, −0.25326714943112172465414344513,
0.25326714943112172465414344513, 0.59975001852631606316816543160, 1.26913325991095883801150110616, 1.39537306277300571392664907869, 1.95685428237061879761503200943, 2.31945273943292824247292950738, 2.83179633898660359022021104389, 3.08399025486773192999829271992, 3.25473597799137279827818728926, 4.01559060560702687459658896699, 4.08227951452974919316051352725, 4.34153736690433087421148271903, 4.55477257525123512930064450708, 4.66245048605440427534088636445, 4.93496312411866146404560896215, 5.36270397000094329740400296419, 5.42033306434568974412527810687, 6.00735878307250279328760243835, 6.28862521698114964792106375290, 6.41138665147779275924755758272, 6.56566879773725328895352107335, 6.71455492544639511164609880072, 7.20663605520152693178792292834, 7.39663048840699601049389016299, 7.49696345504777717971592940945
