| L(s) = 1 | + 2.97·2-s + 0.860·4-s − 24.9·7-s − 21.2·8-s − 63.2·11-s − 90.4·13-s − 74.3·14-s − 70.1·16-s − 10.2·17-s + 24.7·19-s − 188.·22-s − 2.69·23-s − 269.·26-s − 21.4·28-s − 109.·29-s − 69.4·31-s − 38.7·32-s − 30.4·34-s − 23.0·37-s + 73.6·38-s + 266.·41-s + 470.·43-s − 54.4·44-s − 8.01·46-s + 437.·47-s + 280.·49-s − 77.8·52-s + ⋯ |
| L(s) = 1 | + 1.05·2-s + 0.107·4-s − 1.34·7-s − 0.939·8-s − 1.73·11-s − 1.93·13-s − 1.41·14-s − 1.09·16-s − 0.145·17-s + 0.298·19-s − 1.82·22-s − 0.0244·23-s − 2.03·26-s − 0.144·28-s − 0.702·29-s − 0.402·31-s − 0.214·32-s − 0.153·34-s − 0.102·37-s + 0.314·38-s + 1.01·41-s + 1.66·43-s − 0.186·44-s − 0.0256·46-s + 1.35·47-s + 0.817·49-s − 0.207·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4741769506\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4741769506\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 2.97T + 8T^{2} \) |
| 7 | \( 1 + 24.9T + 343T^{2} \) |
| 11 | \( 1 + 63.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 90.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 10.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 24.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 2.69T + 1.21e4T^{2} \) |
| 29 | \( 1 + 109.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 69.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 23.0T + 5.06e4T^{2} \) |
| 41 | \( 1 - 266.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 470.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 437.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 114.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 474.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 333.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 76.1T + 3.00e5T^{2} \) |
| 71 | \( 1 + 658.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 549.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 171.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 332.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 621.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.51e3T + 9.12e5T^{2} \) |
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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
−9.022512280839115699593549104087, −7.66723475958326017391939791488, −7.25269199436005215281284680298, −6.11687900279083771683512313997, −5.49328477508169279973259059508, −4.81234308902464549505652323838, −3.91842183299328232541342250373, −2.77945626055794141924256988086, −2.56672667498395612721755499306, −0.24478374271573570800294830390,
0.24478374271573570800294830390, 2.56672667498395612721755499306, 2.77945626055794141924256988086, 3.91842183299328232541342250373, 4.81234308902464549505652323838, 5.49328477508169279973259059508, 6.11687900279083771683512313997, 7.25269199436005215281284680298, 7.66723475958326017391939791488, 9.022512280839115699593549104087
