| L(s) = 1 | + 2.52·2-s − 3-s + 4.39·4-s − 2.52·6-s + 4.92·7-s + 6.05·8-s + 9-s − 1.13·11-s − 4.39·12-s − 4·13-s + 12.4·14-s + 6.52·16-s + 6.79·17-s + 2.52·18-s − 19-s − 4.92·21-s − 2.86·22-s − 1.92·23-s − 6.05·24-s − 10.1·26-s − 27-s + 21.6·28-s − 5·29-s + 5.13·31-s + 4.39·32-s + 1.13·33-s + 17.1·34-s + ⋯ |
| L(s) = 1 | + 1.78·2-s − 0.577·3-s + 2.19·4-s − 1.03·6-s + 1.86·7-s + 2.14·8-s + 0.333·9-s − 0.341·11-s − 1.26·12-s − 1.10·13-s + 3.32·14-s + 1.63·16-s + 1.64·17-s + 0.596·18-s − 0.229·19-s − 1.07·21-s − 0.611·22-s − 0.401·23-s − 1.23·24-s − 1.98·26-s − 0.192·27-s + 4.09·28-s − 0.928·29-s + 0.922·31-s + 0.777·32-s + 0.197·33-s + 2.94·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.865272408\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.865272408\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 2.52T + 2T^{2} \) |
| 7 | \( 1 - 4.92T + 7T^{2} \) |
| 11 | \( 1 + 1.13T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 - 6.79T + 17T^{2} \) |
| 23 | \( 1 + 1.92T + 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 - 5.13T + 31T^{2} \) |
| 37 | \( 1 + 9.05T + 37T^{2} \) |
| 41 | \( 1 - 3.86T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 4.26T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 - 9.19T + 59T^{2} \) |
| 61 | \( 1 - 0.733T + 61T^{2} \) |
| 67 | \( 1 + 4.86T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 + 0.866T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 - 9.32T + 97T^{2} \) |
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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
−9.989223260868235375363932385683, −8.387353669502539239257027717848, −7.57215368575894029225126272950, −7.00541146682420557332090771025, −5.67152466362370697840996444150, −5.33101823238736231735751297204, −4.66870305760039868359795825944, −3.83758430565656997567348317645, −2.54311255010881289997729599166, −1.54543612997203441825260157120,
1.54543612997203441825260157120, 2.54311255010881289997729599166, 3.83758430565656997567348317645, 4.66870305760039868359795825944, 5.33101823238736231735751297204, 5.67152466362370697840996444150, 7.00541146682420557332090771025, 7.57215368575894029225126272950, 8.387353669502539239257027717848, 9.989223260868235375363932385683
