L(s) = 1 | − 3-s − 5-s + 9-s − 4·11-s + 2·13-s + 15-s − 2·17-s − 8·19-s − 4·23-s + 25-s − 27-s − 6·29-s + 4·33-s + 2·37-s − 2·39-s − 6·41-s − 4·43-s − 45-s + 12·47-s − 7·49-s + 2·51-s − 6·53-s + 4·55-s + 8·57-s − 12·59-s + 14·61-s − 2·65-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.258·15-s − 0.485·17-s − 1.83·19-s − 0.834·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.696·33-s + 0.328·37-s − 0.320·39-s − 0.937·41-s − 0.609·43-s − 0.149·45-s + 1.75·47-s − 49-s + 0.280·51-s − 0.824·53-s + 0.539·55-s + 1.05·57-s − 1.56·59-s + 1.79·61-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−10.79747527742418959252409710261, −9.835497345788981497880663004356, −8.582683577762419953757792101141, −7.888892617242428206551037061144, −6.76366111244607068879792174831, −5.86487938441035862983502527784, −4.78210199118180492802537017050, −3.76484405045814108002996889467, −2.15654793577695836540091578775, 0,
2.15654793577695836540091578775, 3.76484405045814108002996889467, 4.78210199118180492802537017050, 5.86487938441035862983502527784, 6.76366111244607068879792174831, 7.888892617242428206551037061144, 8.582683577762419953757792101141, 9.835497345788981497880663004356, 10.79747527742418959252409710261