| L(s) = 1 | − 2·9-s − 25-s + 4·41-s − 2·49-s + 3·81-s + 4·89-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 2·225-s + ⋯ |
| L(s) = 1 | − 2·9-s − 25-s + 4·41-s − 2·49-s + 3·81-s + 4·89-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 2·225-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7904335686\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7904335686\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_1$ | \( ( 1 - T )^{4} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_1$ | \( ( 1 - T )^{4} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−9.994793330505091035249915327841, −9.482603458082583483086456790282, −9.316453262918297456476663793718, −8.869293515343342907580846639575, −8.480920221106173447274895925969, −8.026880209641628523569417145420, −7.57727930770357627521406680542, −7.54289146383642021304464295970, −6.59957908161261979085605633843, −6.22696551965542421121709836129, −5.92421089632189173095394856323, −5.63658752206409482313247284201, −4.97860869551230558678845950276, −4.64458366134901013714509863897, −3.94044137499837100849588764277, −3.48531571932710923668761114259, −2.93826207171072861338336841391, −2.45287994393897525826934181591, −1.94926018263079764257266318814, −0.75773930440641334800691844760,
0.75773930440641334800691844760, 1.94926018263079764257266318814, 2.45287994393897525826934181591, 2.93826207171072861338336841391, 3.48531571932710923668761114259, 3.94044137499837100849588764277, 4.64458366134901013714509863897, 4.97860869551230558678845950276, 5.63658752206409482313247284201, 5.92421089632189173095394856323, 6.22696551965542421121709836129, 6.59957908161261979085605633843, 7.54289146383642021304464295970, 7.57727930770357627521406680542, 8.026880209641628523569417145420, 8.480920221106173447274895925969, 8.869293515343342907580846639575, 9.316453262918297456476663793718, 9.482603458082583483086456790282, 9.994793330505091035249915327841
