| L(s) = 1 | − 4·5-s − 6·9-s + 12·13-s + 4·17-s + 2·25-s − 20·29-s − 4·37-s + 20·41-s + 24·45-s − 14·49-s + 28·53-s − 20·61-s − 48·65-s − 12·73-s + 27·81-s − 16·85-s + 20·89-s + 36·97-s − 4·101-s + 12·109-s − 28·113-s − 72·117-s − 22·121-s + 28·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 2·9-s + 3.32·13-s + 0.970·17-s + 2/5·25-s − 3.71·29-s − 0.657·37-s + 3.12·41-s + 3.57·45-s − 2·49-s + 3.84·53-s − 2.56·61-s − 5.95·65-s − 1.40·73-s + 3·81-s − 1.73·85-s + 2.11·89-s + 3.65·97-s − 0.398·101-s + 1.14·109-s − 2.63·113-s − 6.65·117-s − 2·121-s + 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4296991136\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4296991136\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
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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
−20.20845899367360484082674298843, −19.18872457828343982814861254091, −19.18872457828343982814861254091, −18.11836294614508901336437330214, −18.11836294614508901336437330214, −16.73856366990991880815681244917, −16.73856366990991880815681244917, −15.74882074786535250024495459074, −15.74882074786535250024495459074, −14.57652563978276046331230069557, −14.57652563978276046331230069557, −13.27687552535142704095990346642, −13.27687552535142704095990346642, −11.76661268274493420855693255102, −11.76661268274493420855693255102, −10.90769214371221130983350005998, −10.90769214371221130983350005998, −8.955386231165229198073332132052, −8.955386231165229198073332132052, −7.77199473906097062385997282225, −7.77199473906097062385997282225, −5.87146418848833687506982135026, −5.87146418848833687506982135026, −3.67478222653086463350186782835, −3.67478222653086463350186782835,
3.67478222653086463350186782835, 3.67478222653086463350186782835, 5.87146418848833687506982135026, 5.87146418848833687506982135026, 7.77199473906097062385997282225, 7.77199473906097062385997282225, 8.955386231165229198073332132052, 8.955386231165229198073332132052, 10.90769214371221130983350005998, 10.90769214371221130983350005998, 11.76661268274493420855693255102, 11.76661268274493420855693255102, 13.27687552535142704095990346642, 13.27687552535142704095990346642, 14.57652563978276046331230069557, 14.57652563978276046331230069557, 15.74882074786535250024495459074, 15.74882074786535250024495459074, 16.73856366990991880815681244917, 16.73856366990991880815681244917, 18.11836294614508901336437330214, 18.11836294614508901336437330214, 19.18872457828343982814861254091, 19.18872457828343982814861254091, 20.20845899367360484082674298843
