L(s) = 1 | − 1.75·3-s + 5-s + 7-s + 0.0856·9-s + 3.23·11-s + 13-s − 1.75·15-s − 4.59·17-s − 0.150·19-s − 1.75·21-s − 1.40·23-s + 25-s + 5.11·27-s − 8.67·29-s + 4.96·31-s − 5.68·33-s + 35-s − 0.520·37-s − 1.75·39-s + 3.55·41-s − 0.277·43-s + 0.0856·45-s + 11.1·47-s + 49-s + 8.07·51-s + 0.171·53-s + 3.23·55-s + ⋯ |
L(s) = 1 | − 1.01·3-s + 0.447·5-s + 0.377·7-s + 0.0285·9-s + 0.975·11-s + 0.277·13-s − 0.453·15-s − 1.11·17-s − 0.0345·19-s − 0.383·21-s − 0.293·23-s + 0.200·25-s + 0.985·27-s − 1.61·29-s + 0.892·31-s − 0.989·33-s + 0.169·35-s − 0.0855·37-s − 0.281·39-s + 0.555·41-s − 0.0422·43-s + 0.0127·45-s + 1.62·47-s + 0.142·49-s + 1.13·51-s + 0.0235·53-s + 0.436·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.430409550\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.430409550\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 + 1.75T + 3T^{2} \) |
| 11 | \( 1 - 3.23T + 11T^{2} \) |
| 17 | \( 1 + 4.59T + 17T^{2} \) |
| 19 | \( 1 + 0.150T + 19T^{2} \) |
| 23 | \( 1 + 1.40T + 23T^{2} \) |
| 29 | \( 1 + 8.67T + 29T^{2} \) |
| 31 | \( 1 - 4.96T + 31T^{2} \) |
| 37 | \( 1 + 0.520T + 37T^{2} \) |
| 41 | \( 1 - 3.55T + 41T^{2} \) |
| 43 | \( 1 + 0.277T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 - 0.171T + 53T^{2} \) |
| 59 | \( 1 + 7.78T + 59T^{2} \) |
| 61 | \( 1 - 3.68T + 61T^{2} \) |
| 67 | \( 1 - 1.95T + 67T^{2} \) |
| 71 | \( 1 - 1.55T + 71T^{2} \) |
| 73 | \( 1 - 5.85T + 73T^{2} \) |
| 79 | \( 1 - 5.00T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 + 15.4T + 89T^{2} \) |
| 97 | \( 1 - 0.869T + 97T^{2} \) |
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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
−7.85190465550338289185586704419, −7.00378507776190600486448976834, −6.37408546761096288302191400046, −5.89862585461653554732379443550, −5.20722845449709909896995106431, −4.43209229818894229918809904366, −3.76507176995003408692302538479, −2.55468273593609552032433178998, −1.67824579450284649120424654375, −0.65019666112248090500848364205,
0.65019666112248090500848364205, 1.67824579450284649120424654375, 2.55468273593609552032433178998, 3.76507176995003408692302538479, 4.43209229818894229918809904366, 5.20722845449709909896995106431, 5.89862585461653554732379443550, 6.37408546761096288302191400046, 7.00378507776190600486448976834, 7.85190465550338289185586704419