| L(s) = 1 | − 5-s − 2·7-s − 4·11-s + 6·13-s − 2·17-s − 8·19-s + 6·23-s + 25-s − 2·29-s + 4·31-s + 2·35-s − 2·37-s + 10·41-s + 2·43-s + 2·47-s − 3·49-s + 2·53-s + 4·55-s − 2·61-s − 6·65-s + 6·67-s + 12·71-s + 10·73-s + 8·77-s − 8·79-s − 10·83-s + 2·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s − 1.20·11-s + 1.66·13-s − 0.485·17-s − 1.83·19-s + 1.25·23-s + 1/5·25-s − 0.371·29-s + 0.718·31-s + 0.338·35-s − 0.328·37-s + 1.56·41-s + 0.304·43-s + 0.291·47-s − 3/7·49-s + 0.274·53-s + 0.539·55-s − 0.256·61-s − 0.744·65-s + 0.733·67-s + 1.42·71-s + 1.17·73-s + 0.911·77-s − 0.900·79-s − 1.09·83-s + 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.229461141\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.229461141\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.637588809241276031769349149702, −8.198425204155087916449678449322, −7.22523714288498578586167895314, −6.43994745928577778528949953628, −5.87904549724101637062073659041, −4.79631332498560161768109396865, −3.97804609056765892400553483994, −3.15160418737570888427061443506, −2.21397268061529340725377684972, −0.66259622561650473080382665441,
0.66259622561650473080382665441, 2.21397268061529340725377684972, 3.15160418737570888427061443506, 3.97804609056765892400553483994, 4.79631332498560161768109396865, 5.87904549724101637062073659041, 6.43994745928577778528949953628, 7.22523714288498578586167895314, 8.198425204155087916449678449322, 8.637588809241276031769349149702
