| L(s) = 1 | + 4·5-s − 3·9-s − 4·13-s + 20·17-s − 38·25-s + 52·29-s − 52·37-s + 116·41-s − 12·45-s + 50·49-s + 148·53-s − 52·61-s − 16·65-s − 92·73-s + 9·81-s + 80·85-s + 164·89-s + 4·97-s + 148·101-s + 92·109-s − 220·113-s + 12·117-s + 194·121-s − 268·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 4/5·5-s − 1/3·9-s − 0.307·13-s + 1.17·17-s − 1.51·25-s + 1.79·29-s − 1.40·37-s + 2.82·41-s − 0.266·45-s + 1.02·49-s + 2.79·53-s − 0.852·61-s − 0.246·65-s − 1.26·73-s + 1/9·81-s + 0.941·85-s + 1.84·89-s + 4/97·97-s + 1.46·101-s + 0.844·109-s − 1.94·113-s + 4/39·117-s + 1.60·121-s − 2.14·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.170071221\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.170071221\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 50 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 194 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 290 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 290 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 1874 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 26 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 58 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 1346 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 382 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 74 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 1150 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 26 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 8930 T^{2} + p^{4} T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 46 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 1390 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 11426 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 82 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.30994236711842367607352152945, −12.14645749425567440806275740897, −11.75708638393646185810657213220, −11.03401133810739354441756124658, −10.34026127358801225949876956283, −10.21374067539048479110659762489, −9.677196034768099341392142727470, −9.003237493038277317007575898471, −8.717165311987638300053998517589, −7.87472050326695064117571857597, −7.53248001597013587796042673047, −6.90152327176605236378331248794, −5.98611054186061833859331948800, −5.86724718930888547116028605221, −5.20911113389778063738607086778, −4.39567593106682269030443981156, −3.68636738717169391107319437857, −2.76962669435860102907115344980, −2.10540305371607791461657824528, −0.899907469215992686316824256879,
0.899907469215992686316824256879, 2.10540305371607791461657824528, 2.76962669435860102907115344980, 3.68636738717169391107319437857, 4.39567593106682269030443981156, 5.20911113389778063738607086778, 5.86724718930888547116028605221, 5.98611054186061833859331948800, 6.90152327176605236378331248794, 7.53248001597013587796042673047, 7.87472050326695064117571857597, 8.717165311987638300053998517589, 9.003237493038277317007575898471, 9.677196034768099341392142727470, 10.21374067539048479110659762489, 10.34026127358801225949876956283, 11.03401133810739354441756124658, 11.75708638393646185810657213220, 12.14645749425567440806275740897, 12.30994236711842367607352152945
