| L(s) = 1 | − 3-s − 7-s + 9-s + 3·11-s + 7·13-s − 8·17-s − 5·19-s + 21-s − 23-s − 27-s + 9·29-s − 6·31-s − 3·33-s + 4·37-s − 7·39-s − 9·41-s − 9·43-s + 2·47-s − 6·49-s + 8·51-s − 2·53-s + 5·57-s + 4·61-s − 63-s + 8·67-s + 69-s + 10·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.904·11-s + 1.94·13-s − 1.94·17-s − 1.14·19-s + 0.218·21-s − 0.208·23-s − 0.192·27-s + 1.67·29-s − 1.07·31-s − 0.522·33-s + 0.657·37-s − 1.12·39-s − 1.40·41-s − 1.37·43-s + 0.291·47-s − 6/7·49-s + 1.12·51-s − 0.274·53-s + 0.662·57-s + 0.512·61-s − 0.125·63-s + 0.977·67-s + 0.120·69-s + 1.18·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−13.73182536601669, −13.34183651233535, −12.96093234659046, −12.55494751073723, −11.77712660775495, −11.39616103264263, −11.10103170763447, −10.55253010646837, −10.10343250993728, −9.464715166392310, −8.788996052072440, −8.574994546992738, −8.199952438516306, −7.137002770150212, −6.670576098796267, −6.334527895502599, −6.136928059192348, −5.258403268148356, −4.581605850429903, −4.169720247905147, −3.644795150434297, −3.047603411554237, −2.043363478470183, −1.642960194449752, −0.8040671118205806, 0,
0.8040671118205806, 1.642960194449752, 2.043363478470183, 3.047603411554237, 3.644795150434297, 4.169720247905147, 4.581605850429903, 5.258403268148356, 6.136928059192348, 6.334527895502599, 6.670576098796267, 7.137002770150212, 8.199952438516306, 8.574994546992738, 8.788996052072440, 9.464715166392310, 10.10343250993728, 10.55253010646837, 11.10103170763447, 11.39616103264263, 11.77712660775495, 12.55494751073723, 12.96093234659046, 13.34183651233535, 13.73182536601669
