| L(s) = 1 | − 3-s + 7-s + 9-s + 4·11-s + 6·13-s − 2·17-s − 6·19-s − 21-s + 2·23-s − 27-s + 6·29-s + 2·31-s − 4·33-s + 4·37-s − 6·39-s + 8·41-s − 4·43-s + 4·47-s + 49-s + 2·51-s − 6·53-s + 6·57-s − 4·59-s + 14·61-s + 63-s + 4·67-s − 2·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s + 1.20·11-s + 1.66·13-s − 0.485·17-s − 1.37·19-s − 0.218·21-s + 0.417·23-s − 0.192·27-s + 1.11·29-s + 0.359·31-s − 0.696·33-s + 0.657·37-s − 0.960·39-s + 1.24·41-s − 0.609·43-s + 0.583·47-s + 1/7·49-s + 0.280·51-s − 0.824·53-s + 0.794·57-s − 0.520·59-s + 1.79·61-s + 0.125·63-s + 0.488·67-s − 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.171291006\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.171291006\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.87957523653164095240330299234, −6.78115940454873209613543683487, −6.44106087285041274763299305234, −5.93218341320336155339175386164, −4.95086842299283317963658859953, −4.18961336482850957867174170085, −3.79239170375933517371421354189, −2.58360155128555286896146625068, −1.53583946524727325708652314966, −0.817175779758193061906800977327,
0.817175779758193061906800977327, 1.53583946524727325708652314966, 2.58360155128555286896146625068, 3.79239170375933517371421354189, 4.18961336482850957867174170085, 4.95086842299283317963658859953, 5.93218341320336155339175386164, 6.44106087285041274763299305234, 6.78115940454873209613543683487, 7.87957523653164095240330299234
