L(s) = 1 | − 3-s − 5-s + 7-s + 9-s + 2·11-s + 15-s − 6·17-s − 4·19-s − 21-s − 23-s + 25-s − 27-s + 2·29-s − 2·31-s − 2·33-s − 35-s − 2·41-s − 4·43-s − 45-s + 4·47-s + 49-s + 6·51-s − 2·55-s + 4·57-s − 6·59-s + 6·61-s + 63-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.603·11-s + 0.258·15-s − 1.45·17-s − 0.917·19-s − 0.218·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s − 0.359·31-s − 0.348·33-s − 0.169·35-s − 0.312·41-s − 0.609·43-s − 0.149·45-s + 0.583·47-s + 1/7·49-s + 0.840·51-s − 0.269·55-s + 0.529·57-s − 0.781·59-s + 0.768·61-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9094990283\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9094990283\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 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
−14.86122687901980, −14.46147154481354, −13.64052375512800, −13.30751009581950, −12.66381467897457, −12.13188962466053, −11.62566487716618, −11.26894534748413, −10.59961732973673, −10.34651224551546, −9.467022833241082, −8.822926707495043, −8.587243804967977, −7.759648128285094, −7.233412450757044, −6.560811155894755, −6.262399727622107, −5.480079489216698, −4.732175952807526, −4.330955673968690, −3.812912174501360, −2.895006193206874, −2.063721485288128, −1.423296046317474, −0.3664419061880140,
0.3664419061880140, 1.423296046317474, 2.063721485288128, 2.895006193206874, 3.812912174501360, 4.330955673968690, 4.732175952807526, 5.480079489216698, 6.262399727622107, 6.560811155894755, 7.233412450757044, 7.759648128285094, 8.587243804967977, 8.822926707495043, 9.467022833241082, 10.34651224551546, 10.59961732973673, 11.26894534748413, 11.62566487716618, 12.13188962466053, 12.66381467897457, 13.30751009581950, 13.64052375512800, 14.46147154481354, 14.86122687901980