| L(s) = 1 | − 9-s − 4·19-s − 12·29-s − 8·31-s + 12·41-s − 2·49-s + 12·59-s − 20·61-s − 12·71-s + 20·79-s + 81-s − 12·89-s − 12·101-s − 4·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 4·171-s + 173-s + ⋯ |
| L(s) = 1 | − 1/3·9-s − 0.917·19-s − 2.22·29-s − 1.43·31-s + 1.87·41-s − 2/7·49-s + 1.56·59-s − 2.56·61-s − 1.42·71-s + 2.25·79-s + 1/9·81-s − 1.27·89-s − 1.19·101-s − 0.383·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.305·171-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47610000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47610000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5701159516\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5701159516\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 23 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−7.894618990271258838482362332938, −7.84325177004026748885720684217, −7.51099364120308993359537574774, −7.14572339449530757266997410522, −6.74370874392025267337939275358, −6.30772265812298919956730078128, −6.09077571627175807068386095491, −5.58304154348854151435419205529, −5.38202517201003626289371329611, −5.11172777273964489799533236863, −4.28975302144482525278030970395, −4.26983774652200597141317345311, −3.84621089162059485664098388963, −3.40207338858494753362436711250, −2.91972584270743481853768013241, −2.53416881449374513532073560176, −1.92236265836866100869575257903, −1.75652702117879535468281759557, −1.02316651823771868751128250463, −0.19586715114141976357915167741,
0.19586715114141976357915167741, 1.02316651823771868751128250463, 1.75652702117879535468281759557, 1.92236265836866100869575257903, 2.53416881449374513532073560176, 2.91972584270743481853768013241, 3.40207338858494753362436711250, 3.84621089162059485664098388963, 4.26983774652200597141317345311, 4.28975302144482525278030970395, 5.11172777273964489799533236863, 5.38202517201003626289371329611, 5.58304154348854151435419205529, 6.09077571627175807068386095491, 6.30772265812298919956730078128, 6.74370874392025267337939275358, 7.14572339449530757266997410522, 7.51099364120308993359537574774, 7.84325177004026748885720684217, 7.894618990271258838482362332938
