| L(s) = 1 | − 8·11-s − 16·19-s − 12·29-s + 12·41-s + 14·49-s + 24·59-s + 28·61-s + 16·79-s + 4·89-s + 28·101-s − 28·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 2.41·11-s − 3.67·19-s − 2.22·29-s + 1.87·41-s + 2·49-s + 3.12·59-s + 3.58·61-s + 1.80·79-s + 0.423·89-s + 2.78·101-s − 2.68·109-s + 2.36·121-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 + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.621680815\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.621680815\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 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 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−8.052824808533117022898045536928, −7.906287554267644760110990007568, −7.29609051134082800358814113612, −7.20014264183481599696242855430, −6.69686101872249739436284130494, −6.41182579100004697809113587017, −5.80333976473364281720755581957, −5.72655344255255830159392598240, −5.20960803097847799986686464430, −5.15410846111931478084596954858, −4.33990479946822647734133775935, −4.24349008523421862213581816283, −3.77028558857676281113920748309, −3.54408379319472328829457109447, −2.54752033829613595268143483912, −2.45678948286317846682402887910, −2.21019137744819749006051435412, −1.88910809898938296870053387701, −0.63749876663459055244269537054, −0.46617264183957075442671633176,
0.46617264183957075442671633176, 0.63749876663459055244269537054, 1.88910809898938296870053387701, 2.21019137744819749006051435412, 2.45678948286317846682402887910, 2.54752033829613595268143483912, 3.54408379319472328829457109447, 3.77028558857676281113920748309, 4.24349008523421862213581816283, 4.33990479946822647734133775935, 5.15410846111931478084596954858, 5.20960803097847799986686464430, 5.72655344255255830159392598240, 5.80333976473364281720755581957, 6.41182579100004697809113587017, 6.69686101872249739436284130494, 7.20014264183481599696242855430, 7.29609051134082800358814113612, 7.906287554267644760110990007568, 8.052824808533117022898045536928
