L(s) = 1 | + 2-s + 2.16·3-s + 4-s − 5-s + 2.16·6-s + 7-s + 8-s + 1.67·9-s − 10-s + 2.16·12-s − 3.92·13-s + 14-s − 2.16·15-s + 16-s − 6.06·17-s + 1.67·18-s − 5.77·19-s − 20-s + 2.16·21-s + 1.51·23-s + 2.16·24-s + 25-s − 3.92·26-s − 2.85·27-s + 28-s − 4.12·29-s − 2.16·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.24·3-s + 0.5·4-s − 0.447·5-s + 0.883·6-s + 0.377·7-s + 0.353·8-s + 0.559·9-s − 0.316·10-s + 0.624·12-s − 1.08·13-s + 0.267·14-s − 0.558·15-s + 0.250·16-s − 1.46·17-s + 0.395·18-s − 1.32·19-s − 0.223·20-s + 0.472·21-s + 0.315·23-s + 0.441·24-s + 0.200·25-s − 0.768·26-s − 0.549·27-s + 0.188·28-s − 0.765·29-s − 0.394·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 - 2.16T + 3T^{2} \) |
| 13 | \( 1 + 3.92T + 13T^{2} \) |
| 17 | \( 1 + 6.06T + 17T^{2} \) |
| 19 | \( 1 + 5.77T + 19T^{2} \) |
| 23 | \( 1 - 1.51T + 23T^{2} \) |
| 29 | \( 1 + 4.12T + 29T^{2} \) |
| 31 | \( 1 - 5.77T + 31T^{2} \) |
| 37 | \( 1 + 4.55T + 37T^{2} \) |
| 41 | \( 1 + 5.54T + 41T^{2} \) |
| 43 | \( 1 - 6.51T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 - 3.74T + 53T^{2} \) |
| 59 | \( 1 + 3.77T + 59T^{2} \) |
| 61 | \( 1 + 0.532T + 61T^{2} \) |
| 67 | \( 1 + 1.15T + 67T^{2} \) |
| 71 | \( 1 - 0.623T + 71T^{2} \) |
| 73 | \( 1 - 3.10T + 73T^{2} \) |
| 79 | \( 1 - 3.76T + 79T^{2} \) |
| 83 | \( 1 + 12.3T + 83T^{2} \) |
| 89 | \( 1 - 3.59T + 89T^{2} \) |
| 97 | \( 1 + 15.6T + 97T^{2} \) |
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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
−7.37535668303322064249956052852, −6.91548140592868209472038671668, −6.13147738618648311913108579757, −5.07508238351369037109191873153, −4.48707477359908688435348497577, −3.92486906127401289644781623735, −3.03935349662536666867092087927, −2.37934078168086316022645722123, −1.78987209666502174251391932816, 0,
1.78987209666502174251391932816, 2.37934078168086316022645722123, 3.03935349662536666867092087927, 3.92486906127401289644781623735, 4.48707477359908688435348497577, 5.07508238351369037109191873153, 6.13147738618648311913108579757, 6.91548140592868209472038671668, 7.37535668303322064249956052852