| L(s) = 1 | − 2·2-s − 1.24·3-s + 4·4-s − 20.5·5-s + 2.49·6-s − 21.2·7-s − 8·8-s − 25.4·9-s + 41.0·10-s − 53.0·11-s − 4.98·12-s + 42.5·14-s + 25.5·15-s + 16·16-s − 69.2·17-s + 50.8·18-s + 46.0·19-s − 82.0·20-s + 26.4·21-s + 106.·22-s − 87.2·23-s + 9.96·24-s + 295.·25-s + 65.3·27-s − 85.0·28-s − 162.·29-s − 51.0·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.239·3-s + 0.5·4-s − 1.83·5-s + 0.169·6-s − 1.14·7-s − 0.353·8-s − 0.942·9-s + 1.29·10-s − 1.45·11-s − 0.119·12-s + 0.811·14-s + 0.439·15-s + 0.250·16-s − 0.987·17-s + 0.666·18-s + 0.556·19-s − 0.916·20-s + 0.275·21-s + 1.02·22-s − 0.790·23-s + 0.0847·24-s + 2.36·25-s + 0.465·27-s − 0.573·28-s − 1.03·29-s − 0.310·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.005400272779\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.005400272779\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 1.24T + 27T^{2} \) |
| 5 | \( 1 + 20.5T + 125T^{2} \) |
| 7 | \( 1 + 21.2T + 343T^{2} \) |
| 11 | \( 1 + 53.0T + 1.33e3T^{2} \) |
| 17 | \( 1 + 69.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 46.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 87.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 162.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 28.3T + 2.97e4T^{2} \) |
| 37 | \( 1 + 111.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 84.2T + 6.89e4T^{2} \) |
| 43 | \( 1 + 328.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 63.2T + 1.03e5T^{2} \) |
| 53 | \( 1 - 721.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 819.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 397.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 77.8T + 3.00e5T^{2} \) |
| 71 | \( 1 - 721.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 57.1T + 3.89e5T^{2} \) |
| 79 | \( 1 + 419.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 917.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 378.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 346.T + 9.12e5T^{2} \) |
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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
−11.09824888573068834314053891531, −10.34427401902915872932934933407, −9.101441335462314469931032751667, −8.213217096450312098464020533651, −7.52295273808437852253385750518, −6.52999771447776280798266922222, −5.21900534661863915228072442131, −3.69795549054480763997185745495, −2.75876212911572623472279802795, −0.05115794935971680226565408560,
0.05115794935971680226565408560, 2.75876212911572623472279802795, 3.69795549054480763997185745495, 5.21900534661863915228072442131, 6.52999771447776280798266922222, 7.52295273808437852253385750518, 8.213217096450312098464020533651, 9.101441335462314469931032751667, 10.34427401902915872932934933407, 11.09824888573068834314053891531
