L(s) = 1 | − 2-s + 4-s − 5-s − 2·7-s − 8-s + 10-s + 4·11-s − 13-s + 2·14-s + 16-s − 6·19-s − 20-s − 4·22-s + 6·23-s + 25-s + 26-s − 2·28-s − 4·29-s − 32-s + 2·35-s + 2·37-s + 6·38-s + 40-s − 2·41-s − 4·43-s + 4·44-s − 6·46-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.755·7-s − 0.353·8-s + 0.316·10-s + 1.20·11-s − 0.277·13-s + 0.534·14-s + 1/4·16-s − 1.37·19-s − 0.223·20-s − 0.852·22-s + 1.25·23-s + 1/5·25-s + 0.196·26-s − 0.377·28-s − 0.742·29-s − 0.176·32-s + 0.338·35-s + 0.328·37-s + 0.973·38-s + 0.158·40-s − 0.312·41-s − 0.609·43-s + 0.603·44-s − 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.479045883\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.479045883\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−12.59683123715983, −11.98727689737529, −11.61182044551486, −11.23225099796911, −10.77217707313654, −10.19016590805583, −9.884012908221377, −9.223005808362387, −9.044612879028406, −8.493137794336222, −8.145192737904952, −7.421957225314258, −6.897588818055795, −6.801855267835799, −6.185124564329973, −5.721253689724638, −5.012901131336183, −4.424589585735100, −3.924137470940268, −3.389941293586930, −2.972849388467432, −2.152905802757833, −1.775831414732788, −0.8653096371324141, −0.4523131180315568,
0.4523131180315568, 0.8653096371324141, 1.775831414732788, 2.152905802757833, 2.972849388467432, 3.389941293586930, 3.924137470940268, 4.424589585735100, 5.012901131336183, 5.721253689724638, 6.185124564329973, 6.801855267835799, 6.897588818055795, 7.421957225314258, 8.145192737904952, 8.493137794336222, 9.044612879028406, 9.223005808362387, 9.884012908221377, 10.19016590805583, 10.77217707313654, 11.23225099796911, 11.61182044551486, 11.98727689737529, 12.59683123715983