| L(s) = 1 | − 2·5-s + 4·7-s − 4·11-s − 2·13-s − 6·17-s + 6·19-s − 8·23-s − 25-s + 6·29-s − 2·31-s − 8·35-s − 37-s + 10·43-s + 12·47-s + 9·49-s + 4·53-s + 8·55-s + 4·59-s + 10·61-s + 4·65-s + 4·67-s + 12·71-s − 10·73-s − 16·77-s − 10·79-s + 12·85-s − 2·89-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.51·7-s − 1.20·11-s − 0.554·13-s − 1.45·17-s + 1.37·19-s − 1.66·23-s − 1/5·25-s + 1.11·29-s − 0.359·31-s − 1.35·35-s − 0.164·37-s + 1.52·43-s + 1.75·47-s + 9/7·49-s + 0.549·53-s + 1.07·55-s + 0.520·59-s + 1.28·61-s + 0.496·65-s + 0.488·67-s + 1.42·71-s − 1.17·73-s − 1.82·77-s − 1.12·79-s + 1.30·85-s − 0.211·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.445357472\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.445357472\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 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 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.183658575072356382315755747505, −7.50705023994796954023679752794, −7.13071802841043723119719457038, −5.81770966781667510465241395383, −5.21950944044730324512259744557, −4.45122647259585232606877672426, −3.96546494091075307491785831413, −2.65310737820385006629042401559, −2.02772116954547213624942796128, −0.63227595326986272755242573162,
0.63227595326986272755242573162, 2.02772116954547213624942796128, 2.65310737820385006629042401559, 3.96546494091075307491785831413, 4.45122647259585232606877672426, 5.21950944044730324512259744557, 5.81770966781667510465241395383, 7.13071802841043723119719457038, 7.50705023994796954023679752794, 8.183658575072356382315755747505
