L(s) = 1 | − 3·3-s − 2·7-s + 9·9-s + 22·13-s − 26·19-s + 6·21-s + 25·25-s − 27·27-s − 46·31-s + 26·37-s − 66·39-s + 22·43-s − 45·49-s + 78·57-s − 74·61-s − 18·63-s + 122·67-s + 46·73-s − 75·75-s + 142·79-s + 81·81-s − 44·91-s + 138·93-s + 2·97-s + 194·103-s + 214·109-s − 78·111-s + ⋯ |
L(s) = 1 | − 3-s − 2/7·7-s + 9-s + 1.69·13-s − 1.36·19-s + 2/7·21-s + 25-s − 27-s − 1.48·31-s + 0.702·37-s − 1.69·39-s + 0.511·43-s − 0.918·49-s + 1.36·57-s − 1.21·61-s − 2/7·63-s + 1.82·67-s + 0.630·73-s − 75-s + 1.79·79-s + 81-s − 0.483·91-s + 1.48·93-s + 2/97·97-s + 1.88·103-s + 1.96·109-s − 0.702·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.294134930\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.294134930\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 11 | \( 1 \) |
good | 5 | \( ( 1 - p T )( 1 + p T ) \) |
| 7 | \( 1 + 2 T + p^{2} T^{2} \) |
| 13 | \( 1 - 22 T + p^{2} T^{2} \) |
| 17 | \( ( 1 - p T )( 1 + p T ) \) |
| 19 | \( 1 + 26 T + p^{2} T^{2} \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( 1 + 46 T + p^{2} T^{2} \) |
| 37 | \( 1 - 26 T + p^{2} T^{2} \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( 1 - 22 T + p^{2} T^{2} \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( ( 1 - p T )( 1 + p T ) \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 + 74 T + p^{2} T^{2} \) |
| 67 | \( 1 - 122 T + p^{2} T^{2} \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 - 46 T + p^{2} T^{2} \) |
| 79 | \( 1 - 142 T + p^{2} T^{2} \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( 1 - 2 T + p^{2} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.327948235034849931800611348491, −8.612181547877674083918959005660, −7.64096512560485930418779254248, −6.55745719825766926707549859971, −6.21872583817440867465294605876, −5.26713830158196818719952156491, −4.27816639044754400446392529217, −3.46033733104200946014585124706, −1.88589477031760618492685551187, −0.69760962946878181898835514055,
0.69760962946878181898835514055, 1.88589477031760618492685551187, 3.46033733104200946014585124706, 4.27816639044754400446392529217, 5.26713830158196818719952156491, 6.21872583817440867465294605876, 6.55745719825766926707549859971, 7.64096512560485930418779254248, 8.612181547877674083918959005660, 9.327948235034849931800611348491