L(s) = 1 | + 2-s + 2.64·3-s + 4-s − 1.93·5-s + 2.64·6-s + 1.07·7-s + 8-s + 3.98·9-s − 1.93·10-s + 1.90·11-s + 2.64·12-s − 2.15·13-s + 1.07·14-s − 5.12·15-s + 16-s + 7.78·17-s + 3.98·18-s − 19-s − 1.93·20-s + 2.82·21-s + 1.90·22-s − 0.222·23-s + 2.64·24-s − 1.24·25-s − 2.15·26-s + 2.60·27-s + 1.07·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.52·3-s + 0.5·4-s − 0.867·5-s + 1.07·6-s + 0.404·7-s + 0.353·8-s + 1.32·9-s − 0.613·10-s + 0.575·11-s + 0.762·12-s − 0.598·13-s + 0.286·14-s − 1.32·15-s + 0.250·16-s + 1.88·17-s + 0.939·18-s − 0.229·19-s − 0.433·20-s + 0.617·21-s + 0.406·22-s − 0.0463·23-s + 0.539·24-s − 0.248·25-s − 0.422·26-s + 0.501·27-s + 0.202·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.802239354\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.802239354\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
good | 3 | \( 1 - 2.64T + 3T^{2} \) |
| 5 | \( 1 + 1.93T + 5T^{2} \) |
| 7 | \( 1 - 1.07T + 7T^{2} \) |
| 11 | \( 1 - 1.90T + 11T^{2} \) |
| 13 | \( 1 + 2.15T + 13T^{2} \) |
| 17 | \( 1 - 7.78T + 17T^{2} \) |
| 23 | \( 1 + 0.222T + 23T^{2} \) |
| 29 | \( 1 + 4.69T + 29T^{2} \) |
| 31 | \( 1 - 8.98T + 31T^{2} \) |
| 37 | \( 1 - 4.54T + 37T^{2} \) |
| 41 | \( 1 + 1.79T + 41T^{2} \) |
| 43 | \( 1 - 2.89T + 43T^{2} \) |
| 47 | \( 1 + 1.39T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 - 1.99T + 59T^{2} \) |
| 61 | \( 1 + 3.71T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 - 5.43T + 71T^{2} \) |
| 73 | \( 1 - 5.89T + 73T^{2} \) |
| 79 | \( 1 + 5.16T + 79T^{2} \) |
| 83 | \( 1 + 0.230T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
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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
−7.889248325518905966830903651109, −7.40944127413133836337435405488, −6.58226818484730100637658436003, −5.62645170071654321239777859643, −4.80955905455950789620064131691, −3.98276255951055454429221947601, −3.61694913385100372763863728125, −2.85363001216188285900462641985, −2.09554531354613954340836466654, −1.04618668322711438954940760340,
1.04618668322711438954940760340, 2.09554531354613954340836466654, 2.85363001216188285900462641985, 3.61694913385100372763863728125, 3.98276255951055454429221947601, 4.80955905455950789620064131691, 5.62645170071654321239777859643, 6.58226818484730100637658436003, 7.40944127413133836337435405488, 7.889248325518905966830903651109