| L(s) = 1 | + 2-s − 0.848·3-s + 4-s − 5-s − 0.848·6-s + 7-s + 8-s − 2.28·9-s − 10-s − 0.0601·11-s − 0.848·12-s − 3.29·13-s + 14-s + 0.848·15-s + 16-s − 4.37·17-s − 2.28·18-s − 0.385·19-s − 20-s − 0.848·21-s − 0.0601·22-s + 0.919·23-s − 0.848·24-s + 25-s − 3.29·26-s + 4.48·27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.489·3-s + 0.5·4-s − 0.447·5-s − 0.346·6-s + 0.377·7-s + 0.353·8-s − 0.760·9-s − 0.316·10-s − 0.0181·11-s − 0.244·12-s − 0.912·13-s + 0.267·14-s + 0.219·15-s + 0.250·16-s − 1.06·17-s − 0.537·18-s − 0.0883·19-s − 0.223·20-s − 0.185·21-s − 0.0128·22-s + 0.191·23-s − 0.173·24-s + 0.200·25-s − 0.645·26-s + 0.862·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4970 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4970 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.836952345\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.836952345\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 71 | \( 1 - T \) |
| good | 3 | \( 1 + 0.848T + 3T^{2} \) |
| 11 | \( 1 + 0.0601T + 11T^{2} \) |
| 13 | \( 1 + 3.29T + 13T^{2} \) |
| 17 | \( 1 + 4.37T + 17T^{2} \) |
| 19 | \( 1 + 0.385T + 19T^{2} \) |
| 23 | \( 1 - 0.919T + 23T^{2} \) |
| 29 | \( 1 + 10.1T + 29T^{2} \) |
| 31 | \( 1 - 5.32T + 31T^{2} \) |
| 37 | \( 1 - 2.57T + 37T^{2} \) |
| 41 | \( 1 - 11.9T + 41T^{2} \) |
| 43 | \( 1 - 9.91T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 + 3.23T + 53T^{2} \) |
| 59 | \( 1 - 7.97T + 59T^{2} \) |
| 61 | \( 1 - 3.46T + 61T^{2} \) |
| 67 | \( 1 + 9.42T + 67T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 - 2.91T + 79T^{2} \) |
| 83 | \( 1 - 7.10T + 83T^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 + 12.1T + 97T^{2} \) |
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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
−8.039835701574050239012891788923, −7.44306244289134513756778873766, −6.74985505320893318551304446131, −5.86167300220147953566828530143, −5.39580557829578344310980568252, −4.48510237953148464231921520259, −4.02480865505582380718063672884, −2.79520827110573158245352938662, −2.21383774732375518262349713942, −0.65628846013863345195990449523,
0.65628846013863345195990449523, 2.21383774732375518262349713942, 2.79520827110573158245352938662, 4.02480865505582380718063672884, 4.48510237953148464231921520259, 5.39580557829578344310980568252, 5.86167300220147953566828530143, 6.74985505320893318551304446131, 7.44306244289134513756778873766, 8.039835701574050239012891788923
