L(s) = 1 | + 1.41·2-s + 2.00·4-s + 2.64i·7-s + 2.82·8-s − 2.32i·11-s − 10.6i·13-s + 3.74i·14-s + 4.00·16-s + 2.70·17-s − 11.0·19-s − 3.29i·22-s − 34.7·23-s − 15.1i·26-s + 5.29i·28-s + 5.82i·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.500·4-s + 0.377i·7-s + 0.353·8-s − 0.211i·11-s − 0.822i·13-s + 0.267i·14-s + 0.250·16-s + 0.159·17-s − 0.584·19-s − 0.149i·22-s − 1.50·23-s − 0.581i·26-s + 0.188i·28-s + 0.200i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 + 0.472i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8450955870\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8450955870\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
good | 11 | \( 1 + 2.32iT - 121T^{2} \) |
| 13 | \( 1 + 10.6iT - 169T^{2} \) |
| 17 | \( 1 - 2.70T + 289T^{2} \) |
| 19 | \( 1 + 11.0T + 361T^{2} \) |
| 23 | \( 1 + 34.7T + 529T^{2} \) |
| 29 | \( 1 - 5.82iT - 841T^{2} \) |
| 31 | \( 1 + 15.2T + 961T^{2} \) |
| 37 | \( 1 + 0.900iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 31.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 13.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 17.1T + 2.20e3T^{2} \) |
| 53 | \( 1 + 68.1T + 2.80e3T^{2} \) |
| 59 | \( 1 - 34.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 81.5T + 3.72e3T^{2} \) |
| 67 | \( 1 + 46.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 61.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 20.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 78.0T + 6.24e3T^{2} \) |
| 83 | \( 1 - 28.7T + 6.88e3T^{2} \) |
| 89 | \( 1 - 136. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 10.0iT - 9.40e3T^{2} \) |
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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
−8.064273442616460951501007870448, −7.47758753640553555517860598424, −6.44691364728802975729422432243, −5.85929464044059792297533217122, −5.19623209345829868544619321115, −4.25258908761991570800709404087, −3.46773092971627077639393651715, −2.57999265592651939084649007244, −1.63270069059436567438691257254, −0.12855695618569474469924825182,
1.47351916552568935610673445882, 2.32659589495495637416740722686, 3.44085016437924028146650111465, 4.23418637093014843690359161545, 4.78275086599152392596412745430, 5.90597437137535189793595111271, 6.39873153019901398961271194606, 7.26899858936126858315864660368, 7.909605483084047736038731290337, 8.764010908384338832771773126267