| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s − 6·11-s + 13-s + 14-s + 16-s + 3·17-s − 4·19-s + 6·22-s − 3·23-s − 26-s − 28-s − 3·29-s + 5·31-s − 32-s − 3·34-s + 10·37-s + 4·38-s − 9·41-s + 43-s − 6·44-s + 3·46-s + 49-s + 52-s + 9·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 1.80·11-s + 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.917·19-s + 1.27·22-s − 0.625·23-s − 0.196·26-s − 0.188·28-s − 0.557·29-s + 0.898·31-s − 0.176·32-s − 0.514·34-s + 1.64·37-s + 0.648·38-s − 1.40·41-s + 0.152·43-s − 0.904·44-s + 0.442·46-s + 1/7·49-s + 0.138·52-s + 1.23·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8696976244\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8696976244\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.453884424345087142356650614290, −8.091044136111993741067571884313, −7.38980994272698869605072344429, −6.47517353189684474418126099661, −5.76214448647638297000703306978, −4.97555100712520025613173196251, −3.83517897824153179547742671926, −2.82922405813347082842558391330, −2.09056417572198201018678635137, −0.59867265186005947678509869294,
0.59867265186005947678509869294, 2.09056417572198201018678635137, 2.82922405813347082842558391330, 3.83517897824153179547742671926, 4.97555100712520025613173196251, 5.76214448647638297000703306978, 6.47517353189684474418126099661, 7.38980994272698869605072344429, 8.091044136111993741067571884313, 8.453884424345087142356650614290
