| L(s) = 1 | − 3-s − 4.31·7-s + 9-s − 2.26·11-s − 2.25·13-s − 0.535·17-s − 6.91·19-s + 4.31·21-s + 23-s − 27-s + 5.64·29-s − 10.4·31-s + 2.26·33-s − 8.10·37-s + 2.25·39-s + 0.633·41-s + 1.30·43-s − 12.4·47-s + 11.6·49-s + 0.535·51-s − 5.98·53-s + 6.91·57-s + 7.01·59-s + 5.98·61-s − 4.31·63-s − 6.32·67-s − 69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.63·7-s + 0.333·9-s − 0.683·11-s − 0.625·13-s − 0.129·17-s − 1.58·19-s + 0.941·21-s + 0.208·23-s − 0.192·27-s + 1.04·29-s − 1.87·31-s + 0.394·33-s − 1.33·37-s + 0.361·39-s + 0.0988·41-s + 0.199·43-s − 1.81·47-s + 1.66·49-s + 0.0749·51-s − 0.822·53-s + 0.916·57-s + 0.912·59-s + 0.766·61-s − 0.543·63-s − 0.772·67-s − 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2707934063\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2707934063\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + 4.31T + 7T^{2} \) |
| 11 | \( 1 + 2.26T + 11T^{2} \) |
| 13 | \( 1 + 2.25T + 13T^{2} \) |
| 17 | \( 1 + 0.535T + 17T^{2} \) |
| 19 | \( 1 + 6.91T + 19T^{2} \) |
| 29 | \( 1 - 5.64T + 29T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 + 8.10T + 37T^{2} \) |
| 41 | \( 1 - 0.633T + 41T^{2} \) |
| 43 | \( 1 - 1.30T + 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 + 5.98T + 53T^{2} \) |
| 59 | \( 1 - 7.01T + 59T^{2} \) |
| 61 | \( 1 - 5.98T + 61T^{2} \) |
| 67 | \( 1 + 6.32T + 67T^{2} \) |
| 71 | \( 1 - 0.151T + 71T^{2} \) |
| 73 | \( 1 + 8.88T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 - 0.716T + 83T^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
| 97 | \( 1 + 16.2T + 97T^{2} \) |
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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
−7.88612052724543709580416819087, −6.93928531727648465798578226266, −6.68557067897684192560669365636, −5.90163202763714756536951624981, −5.21882724362290049156659065808, −4.40932556668522227753733358200, −3.53933186663337835067863365111, −2.79630865637392862364791023179, −1.86126475885188767619162993175, −0.25620204226968129123775464905,
0.25620204226968129123775464905, 1.86126475885188767619162993175, 2.79630865637392862364791023179, 3.53933186663337835067863365111, 4.40932556668522227753733358200, 5.21882724362290049156659065808, 5.90163202763714756536951624981, 6.68557067897684192560669365636, 6.93928531727648465798578226266, 7.88612052724543709580416819087
