| L(s) = 1 | + 2-s − 3-s + 4-s − 4.05·5-s − 6-s − 1.59·7-s + 8-s + 9-s − 4.05·10-s − 11-s − 12-s + 3.49·13-s − 1.59·14-s + 4.05·15-s + 16-s + 7.31·17-s + 18-s − 1.76·19-s − 4.05·20-s + 1.59·21-s − 22-s − 6.99·23-s − 24-s + 11.4·25-s + 3.49·26-s − 27-s − 1.59·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.81·5-s − 0.408·6-s − 0.602·7-s + 0.353·8-s + 0.333·9-s − 1.28·10-s − 0.301·11-s − 0.288·12-s + 0.969·13-s − 0.426·14-s + 1.04·15-s + 0.250·16-s + 1.77·17-s + 0.235·18-s − 0.405·19-s − 0.907·20-s + 0.348·21-s − 0.213·22-s − 1.45·23-s − 0.204·24-s + 2.29·25-s + 0.685·26-s − 0.192·27-s − 0.301·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 5 | \( 1 + 4.05T + 5T^{2} \) |
| 7 | \( 1 + 1.59T + 7T^{2} \) |
| 13 | \( 1 - 3.49T + 13T^{2} \) |
| 17 | \( 1 - 7.31T + 17T^{2} \) |
| 19 | \( 1 + 1.76T + 19T^{2} \) |
| 23 | \( 1 + 6.99T + 23T^{2} \) |
| 29 | \( 1 - 0.0241T + 29T^{2} \) |
| 31 | \( 1 - 8.75T + 31T^{2} \) |
| 37 | \( 1 + 4.81T + 37T^{2} \) |
| 41 | \( 1 + 7.72T + 41T^{2} \) |
| 43 | \( 1 - 9.27T + 43T^{2} \) |
| 47 | \( 1 + 2.91T + 47T^{2} \) |
| 53 | \( 1 + 6.04T + 53T^{2} \) |
| 59 | \( 1 + 1.28T + 59T^{2} \) |
| 67 | \( 1 - 7.70T + 67T^{2} \) |
| 71 | \( 1 - 5.90T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 - 17.4T + 83T^{2} \) |
| 89 | \( 1 + 15.3T + 89T^{2} \) |
| 97 | \( 1 + 7.51T + 97T^{2} \) |
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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
−8.093295116041725139205470781276, −7.28027982517926208983886913833, −6.51842212534165675450113601243, −5.86606877603207433692280710661, −4.98657915350026692510641947573, −4.12397396829483378373278269079, −3.62409989859867215843129123480, −2.92001937987053975942732179071, −1.23726867704080035218000680397, 0,
1.23726867704080035218000680397, 2.92001937987053975942732179071, 3.62409989859867215843129123480, 4.12397396829483378373278269079, 4.98657915350026692510641947573, 5.86606877603207433692280710661, 6.51842212534165675450113601243, 7.28027982517926208983886913833, 8.093295116041725139205470781276
