L(s) = 1 | − 3-s + 4·7-s + 9-s + 4·11-s − 13-s − 2·17-s + 8·19-s − 4·21-s − 27-s + 6·29-s + 4·31-s − 4·33-s + 2·37-s + 39-s − 10·41-s + 4·43-s + 8·47-s + 9·49-s + 2·51-s + 10·53-s − 8·57-s − 4·59-s − 2·61-s + 4·63-s − 16·67-s + 8·71-s − 2·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.51·7-s + 1/3·9-s + 1.20·11-s − 0.277·13-s − 0.485·17-s + 1.83·19-s − 0.872·21-s − 0.192·27-s + 1.11·29-s + 0.718·31-s − 0.696·33-s + 0.328·37-s + 0.160·39-s − 1.56·41-s + 0.609·43-s + 1.16·47-s + 9/7·49-s + 0.280·51-s + 1.37·53-s − 1.05·57-s − 0.520·59-s − 0.256·61-s + 0.503·63-s − 1.95·67-s + 0.949·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.846655147\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.846655147\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−16.03020536668995, −15.37754228657386, −14.95133939215020, −14.25630784311151, −13.83778284641212, −13.44960979537991, −12.29397000347378, −11.94610989334890, −11.63997380665377, −11.08911358402335, −10.37000249344876, −9.879081791767015, −8.998626010746727, −8.683367078746478, −7.754696146287711, −7.383110755798420, −6.650738213349462, −5.987426046291940, −5.234307692212765, −4.734579702002488, −4.192721668837997, −3.299165328597083, −2.299643408764512, −1.383851677647705, −0.8712609500282515,
0.8712609500282515, 1.383851677647705, 2.299643408764512, 3.299165328597083, 4.192721668837997, 4.734579702002488, 5.234307692212765, 5.987426046291940, 6.650738213349462, 7.383110755798420, 7.754696146287711, 8.683367078746478, 8.998626010746727, 9.879081791767015, 10.37000249344876, 11.08911358402335, 11.63997380665377, 11.94610989334890, 12.29397000347378, 13.44960979537991, 13.83778284641212, 14.25630784311151, 14.95133939215020, 15.37754228657386, 16.03020536668995