| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s − 13-s + 15-s − 3·17-s + 19-s − 21-s − 6·23-s + 25-s − 27-s + 3·29-s + 4·31-s − 35-s − 37-s + 39-s + 4·43-s − 45-s − 6·47-s − 6·49-s + 3·51-s + 6·53-s − 57-s + 6·59-s − 10·61-s + 63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.277·13-s + 0.258·15-s − 0.727·17-s + 0.229·19-s − 0.218·21-s − 1.25·23-s + 1/5·25-s − 0.192·27-s + 0.557·29-s + 0.718·31-s − 0.169·35-s − 0.164·37-s + 0.160·39-s + 0.609·43-s − 0.149·45-s − 0.875·47-s − 6/7·49-s + 0.420·51-s + 0.824·53-s − 0.132·57-s + 0.781·59-s − 1.28·61-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−15.66166123838191, −14.89216056836532, −14.48199539131920, −13.76331738472025, −13.40682178350372, −12.61807402495277, −12.15647370918830, −11.74182927848859, −11.21133069842995, −10.69552966775721, −10.10147396566855, −9.602189247414716, −8.871945351171085, −8.234485577338447, −7.799606330808023, −7.162104656498170, −6.466228189110522, −6.088399971116731, −5.193180419517382, −4.752132018254764, −4.142736722984458, −3.487066091315940, −2.555969003907213, −1.860770975861024, −0.9041645862580797, 0,
0.9041645862580797, 1.860770975861024, 2.555969003907213, 3.487066091315940, 4.142736722984458, 4.752132018254764, 5.193180419517382, 6.088399971116731, 6.466228189110522, 7.162104656498170, 7.799606330808023, 8.234485577338447, 8.871945351171085, 9.602189247414716, 10.10147396566855, 10.69552966775721, 11.21133069842995, 11.74182927848859, 12.15647370918830, 12.61807402495277, 13.40682178350372, 13.76331738472025, 14.48199539131920, 14.89216056836532, 15.66166123838191
