| L(s) = 1 | + 3·2-s − 5·3-s + 6·4-s + 2·5-s − 15·6-s − 3·7-s + 10·8-s + 10·9-s + 6·10-s + 3·11-s − 30·12-s − 9·14-s − 10·15-s + 15·16-s − 7·17-s + 30·18-s − 11·19-s + 12·20-s + 15·21-s + 9·22-s − 14·23-s − 50·24-s − 3·25-s − 6·27-s − 18·28-s + 2·29-s − 30·30-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 2.88·3-s + 3·4-s + 0.894·5-s − 6.12·6-s − 1.13·7-s + 3.53·8-s + 10/3·9-s + 1.89·10-s + 0.904·11-s − 8.66·12-s − 2.40·14-s − 2.58·15-s + 15/4·16-s − 1.69·17-s + 7.07·18-s − 2.52·19-s + 2.68·20-s + 3.27·21-s + 1.91·22-s − 2.91·23-s − 10.2·24-s − 3/5·25-s − 1.15·27-s − 3.40·28-s + 0.371·29-s − 5.47·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 2 | $C_1$ | \( ( 1 - T )^{3} \) |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) |
| 13 | | \( 1 \) |
| good | 3 | $A_4\times C_2$ | \( 1 + 5 T + 5 p T^{2} + 31 T^{3} + 5 p^{2} T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $A_4\times C_2$ | \( 1 - 2 T + 7 T^{2} - 12 T^{3} + 7 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 - 3 T + 15 T^{2} - 53 T^{3} + 15 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 7 T + 65 T^{2} + 245 T^{3} + 65 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 + 11 T + 53 T^{2} + 207 T^{3} + 53 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 14 T + 125 T^{2} + 700 T^{3} + 125 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 - 2 T + 79 T^{2} - 108 T^{3} + 79 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 14 T + 149 T^{2} + 924 T^{3} + 149 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 8 T + 123 T^{2} + 584 T^{3} + 123 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + 9 T + 59 T^{2} + 5 p T^{3} + 59 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 - 19 T + 219 T^{2} - 1663 T^{3} + 219 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 14 T + 169 T^{2} - 1260 T^{3} + 169 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 14 T + 215 T^{2} + 1540 T^{3} + 215 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 9 T - 13 T^{2} - 745 T^{3} - 13 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 4 T + 123 T^{2} + 256 T^{3} + 123 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 3 T + 141 T^{2} - 151 T^{3} + 141 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 8 T + 169 T^{2} + 792 T^{3} + 169 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 + 13 T + 203 T^{2} + 1731 T^{3} + 203 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 16 T + 201 T^{2} + 2296 T^{3} + 201 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 21 T + 333 T^{2} + 3577 T^{3} + 333 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 25 T + 431 T^{2} + 4547 T^{3} + 431 p T^{4} + 25 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 31 T + 609 T^{2} + 7093 T^{3} + 609 p T^{4} + 31 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.553002730704364656247266730966, −7.85364217211133381955964297298, −7.51793227111772361312841676103, −7.28852276793829001896923781277, −6.93519185599674398704594223247, −6.68191337384785991318303201294, −6.64494291065885444291245960204, −6.14880098932720461896367868909, −6.11914647219122825207415566045, −6.00832734024824466460998077874, −5.63310137788663928691385174312, −5.59077284795149942167951913297, −5.55073308603095031941365600544, −4.90726479932843520709803708219, −4.47472768324780206959237674319, −4.43571609583234830811111107164, −4.08905775562965259187456852663, −3.87473658212911604941944085306, −3.83907833483421840763890757127, −3.07149845718852123283194500632, −2.65406921201735599756584591509, −2.47072324790139470021449292389, −1.95233551606990059219029172941, −1.66190035772317680079072420140, −1.45209516289956490994440747952, 0, 0, 0,
1.45209516289956490994440747952, 1.66190035772317680079072420140, 1.95233551606990059219029172941, 2.47072324790139470021449292389, 2.65406921201735599756584591509, 3.07149845718852123283194500632, 3.83907833483421840763890757127, 3.87473658212911604941944085306, 4.08905775562965259187456852663, 4.43571609583234830811111107164, 4.47472768324780206959237674319, 4.90726479932843520709803708219, 5.55073308603095031941365600544, 5.59077284795149942167951913297, 5.63310137788663928691385174312, 6.00832734024824466460998077874, 6.11914647219122825207415566045, 6.14880098932720461896367868909, 6.64494291065885444291245960204, 6.68191337384785991318303201294, 6.93519185599674398704594223247, 7.28852276793829001896923781277, 7.51793227111772361312841676103, 7.85364217211133381955964297298, 8.553002730704364656247266730966
