| L(s) = 1 | − 5-s − 4·7-s + 2·11-s + 4·13-s + 17-s − 5·19-s + 5·23-s + 25-s + 8·29-s + 7·31-s + 4·35-s − 6·37-s + 6·41-s − 2·43-s + 8·47-s + 9·49-s + 9·53-s − 2·55-s + 4·59-s + 13·61-s − 4·65-s − 10·67-s − 6·71-s − 6·73-s − 8·77-s + 9·79-s − 17·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.51·7-s + 0.603·11-s + 1.10·13-s + 0.242·17-s − 1.14·19-s + 1.04·23-s + 1/5·25-s + 1.48·29-s + 1.25·31-s + 0.676·35-s − 0.986·37-s + 0.937·41-s − 0.304·43-s + 1.16·47-s + 9/7·49-s + 1.23·53-s − 0.269·55-s + 0.520·59-s + 1.66·61-s − 0.496·65-s − 1.22·67-s − 0.712·71-s − 0.702·73-s − 0.911·77-s + 1.01·79-s − 1.86·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.294280097\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.294280097\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 + 17 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−9.989897574082467175592855310048, −8.829441665282381881849283797419, −8.563910368217065269879413774060, −7.15433652246950192729082115922, −6.54059618701542467383965359660, −5.86016508216501546204903717806, −4.43606130296782347300876966431, −3.61879914318796247021019774004, −2.72910564029766129097812481232, −0.882165819690340379597945901815,
0.882165819690340379597945901815, 2.72910564029766129097812481232, 3.61879914318796247021019774004, 4.43606130296782347300876966431, 5.86016508216501546204903717806, 6.54059618701542467383965359660, 7.15433652246950192729082115922, 8.563910368217065269879413774060, 8.829441665282381881849283797419, 9.989897574082467175592855310048
