| L(s) = 1 | + 3·3-s + 7-s + 6·9-s + 5·11-s − 3·13-s + 17-s − 6·19-s + 3·21-s − 6·23-s + 9·27-s + 9·29-s − 4·31-s + 15·33-s + 2·37-s − 9·39-s − 4·41-s + 10·43-s + 47-s + 49-s + 3·51-s + 4·53-s − 18·57-s + 8·59-s + 8·61-s + 6·63-s + 12·67-s − 18·69-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.377·7-s + 2·9-s + 1.50·11-s − 0.832·13-s + 0.242·17-s − 1.37·19-s + 0.654·21-s − 1.25·23-s + 1.73·27-s + 1.67·29-s − 0.718·31-s + 2.61·33-s + 0.328·37-s − 1.44·39-s − 0.624·41-s + 1.52·43-s + 0.145·47-s + 1/7·49-s + 0.420·51-s + 0.549·53-s − 2.38·57-s + 1.04·59-s + 1.02·61-s + 0.755·63-s + 1.46·67-s − 2.16·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.918959684\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.918959684\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−16.27632609556893, −15.85267771173173, −14.96868449915192, −14.71879468402029, −14.28166603689159, −13.88039266325613, −13.21486938622897, −12.31361571751191, −12.23936792559716, −11.28045140083084, −10.45704557408217, −9.860215316688768, −9.369893287877594, −8.758891223923112, −8.249933245911293, −7.813115562190910, −6.892828544685557, −6.578853587552279, −5.505088950681397, −4.397068677088044, −4.098129209940456, −3.408678056646960, −2.344677055360641, −2.068019463504143, −0.9691943992752035,
0.9691943992752035, 2.068019463504143, 2.344677055360641, 3.408678056646960, 4.098129209940456, 4.397068677088044, 5.505088950681397, 6.578853587552279, 6.892828544685557, 7.813115562190910, 8.249933245911293, 8.758891223923112, 9.369893287877594, 9.860215316688768, 10.45704557408217, 11.28045140083084, 12.23936792559716, 12.31361571751191, 13.21486938622897, 13.88039266325613, 14.28166603689159, 14.71879468402029, 14.96868449915192, 15.85267771173173, 16.27632609556893
