L(s) = 1 | + 2-s − 2.41·3-s + 4-s − 2.41·6-s − 1.58·7-s + 8-s + 2.82·9-s + 1.41·11-s − 2.41·12-s − 0.171·13-s − 1.58·14-s + 16-s − 17-s + 2.82·18-s − 19-s + 3.82·21-s + 1.41·22-s − 9.24·23-s − 2.41·24-s − 0.171·26-s + 0.414·27-s − 1.58·28-s − 5.82·29-s − 2.24·31-s + 32-s − 3.41·33-s − 34-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.39·3-s + 0.5·4-s − 0.985·6-s − 0.599·7-s + 0.353·8-s + 0.942·9-s + 0.426·11-s − 0.696·12-s − 0.0475·13-s − 0.423·14-s + 0.250·16-s − 0.242·17-s + 0.666·18-s − 0.229·19-s + 0.835·21-s + 0.301·22-s − 1.92·23-s − 0.492·24-s − 0.0336·26-s + 0.0797·27-s − 0.299·28-s − 1.08·29-s − 0.402·31-s + 0.176·32-s − 0.594·33-s − 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + 2.41T + 3T^{2} \) |
| 7 | \( 1 + 1.58T + 7T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 + 0.171T + 13T^{2} \) |
| 17 | \( 1 + T + 17T^{2} \) |
| 23 | \( 1 + 9.24T + 23T^{2} \) |
| 29 | \( 1 + 5.82T + 29T^{2} \) |
| 31 | \( 1 + 2.24T + 31T^{2} \) |
| 37 | \( 1 + 8.48T + 37T^{2} \) |
| 41 | \( 1 - 4.24T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 + 12.8T + 59T^{2} \) |
| 61 | \( 1 - 5.75T + 61T^{2} \) |
| 67 | \( 1 + 13.2T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 - 5.48T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 - 2.48T + 83T^{2} \) |
| 89 | \( 1 + 7.07T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−9.990882895247461372351482260707, −8.873214855841981064270481712128, −7.60788726526879247062694078980, −6.66322976429905311776263874378, −6.08661881876646631595050138314, −5.38621816385852542083535268864, −4.38463132430675439502205353518, −3.46644205862699798920171307809, −1.85947418805391676913251396052, 0,
1.85947418805391676913251396052, 3.46644205862699798920171307809, 4.38463132430675439502205353518, 5.38621816385852542083535268864, 6.08661881876646631595050138314, 6.66322976429905311776263874378, 7.60788726526879247062694078980, 8.873214855841981064270481712128, 9.990882895247461372351482260707