| L(s) = 1 | + 3-s + 4·7-s + 9-s + 2·13-s + 6·17-s + 4·21-s + 4·23-s + 27-s − 2·29-s − 8·31-s − 6·37-s + 2·39-s − 6·41-s − 12·43-s + 12·47-s + 9·49-s + 6·51-s + 10·53-s + 8·59-s − 10·61-s + 4·63-s + 12·67-s + 4·69-s + 8·71-s − 10·73-s + 16·79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.51·7-s + 1/3·9-s + 0.554·13-s + 1.45·17-s + 0.872·21-s + 0.834·23-s + 0.192·27-s − 0.371·29-s − 1.43·31-s − 0.986·37-s + 0.320·39-s − 0.937·41-s − 1.82·43-s + 1.75·47-s + 9/7·49-s + 0.840·51-s + 1.37·53-s + 1.04·59-s − 1.28·61-s + 0.503·63-s + 1.46·67-s + 0.481·69-s + 0.949·71-s − 1.17·73-s + 1.80·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.916778265\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.916778265\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 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 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−8.705123030777718551062740746738, −8.327275882404076664812739978493, −7.50997107238151608216676165196, −6.92530805420045658384192775774, −5.52811097399805640485701497969, −5.15807146102017226607398116108, −4.01150010595379173537756771542, −3.28316059994551513635790000973, −2.01099980219901421650957341119, −1.20740065177480422433876237100,
1.20740065177480422433876237100, 2.01099980219901421650957341119, 3.28316059994551513635790000973, 4.01150010595379173537756771542, 5.15807146102017226607398116108, 5.52811097399805640485701497969, 6.92530805420045658384192775774, 7.50997107238151608216676165196, 8.327275882404076664812739978493, 8.705123030777718551062740746738
