| L(s) = 1 | + 3-s + 9-s − 4·13-s − 4·17-s + 4·19-s + 4·23-s + 27-s + 2·29-s − 8·31-s + 6·37-s − 4·39-s + 12·41-s + 4·43-s − 8·47-s − 4·51-s − 6·53-s + 4·57-s − 12·59-s + 4·61-s − 4·67-s + 4·69-s + 12·71-s − 8·73-s + 16·79-s + 81-s − 4·83-s + 2·87-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 1.10·13-s − 0.970·17-s + 0.917·19-s + 0.834·23-s + 0.192·27-s + 0.371·29-s − 1.43·31-s + 0.986·37-s − 0.640·39-s + 1.87·41-s + 0.609·43-s − 1.16·47-s − 0.560·51-s − 0.824·53-s + 0.529·57-s − 1.56·59-s + 0.512·61-s − 0.488·67-s + 0.481·69-s + 1.42·71-s − 0.936·73-s + 1.80·79-s + 1/9·81-s − 0.439·83-s + 0.214·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−14.58285395472520, −14.17119119425071, −13.60180822554929, −13.04511093499664, −12.64344732648705, −12.19723301397633, −11.48496907060772, −10.92567276403238, −10.67994884740956, −9.674437777648386, −9.402893341976816, −9.174520247869460, −8.302118046040088, −7.787251973756648, −7.365100367976231, −6.836292930095360, −6.222505602951358, −5.477233695800432, −4.902587307105388, −4.396862354911482, −3.747167601908278, −2.931850576646368, −2.581053786600828, −1.819205281853392, −0.9981022325069598, 0,
0.9981022325069598, 1.819205281853392, 2.581053786600828, 2.931850576646368, 3.747167601908278, 4.396862354911482, 4.902587307105388, 5.477233695800432, 6.222505602951358, 6.836292930095360, 7.365100367976231, 7.787251973756648, 8.302118046040088, 9.174520247869460, 9.402893341976816, 9.674437777648386, 10.67994884740956, 10.92567276403238, 11.48496907060772, 12.19723301397633, 12.64344732648705, 13.04511093499664, 13.60180822554929, 14.17119119425071, 14.58285395472520
