| L(s) = 1 | + 4·5-s − 6·11-s + 11·25-s − 10·29-s − 4·31-s − 4·41-s − 49-s − 24·55-s − 20·59-s − 16·61-s − 16·71-s − 10·79-s − 24·101-s − 10·109-s + 5·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s − 40·145-s + 149-s + 151-s − 16·155-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1.80·11-s + 11/5·25-s − 1.85·29-s − 0.718·31-s − 0.624·41-s − 1/7·49-s − 3.23·55-s − 2.60·59-s − 2.04·61-s − 1.89·71-s − 1.12·79-s − 2.38·101-s − 0.957·109-s + 5/11·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3.32·145-s + 0.0819·149-s + 0.0813·151-s − 1.28·155-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25401600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25401600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9740388442\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9740388442\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 145 T^{2} + p^{2} T^{4} \) |
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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
−8.421276714155027935049271496681, −7.912489607461957234786999479884, −7.81801562583209300430738613440, −7.25487659038475336334310034599, −7.07013675983110137248940051656, −6.57977418550555727768402729703, −6.08411997915710905203906552744, −5.71951791940547777491456889722, −5.69214811450080749688387268999, −5.23687183343270890006493926203, −4.87092409289783236742690047893, −4.45551606455866937710485703593, −4.03654460679869530761222033307, −3.15275937995037669846998209216, −3.07206289822361577740838335594, −2.68435576743602893656661144546, −2.00946959593697493241425123315, −1.73785184604692425663931569000, −1.38153444030830393746213131856, −0.24005850478639178493711105714,
0.24005850478639178493711105714, 1.38153444030830393746213131856, 1.73785184604692425663931569000, 2.00946959593697493241425123315, 2.68435576743602893656661144546, 3.07206289822361577740838335594, 3.15275937995037669846998209216, 4.03654460679869530761222033307, 4.45551606455866937710485703593, 4.87092409289783236742690047893, 5.23687183343270890006493926203, 5.69214811450080749688387268999, 5.71951791940547777491456889722, 6.08411997915710905203906552744, 6.57977418550555727768402729703, 7.07013675983110137248940051656, 7.25487659038475336334310034599, 7.81801562583209300430738613440, 7.912489607461957234786999479884, 8.421276714155027935049271496681
