| L(s) = 1 | − 1.52·3-s − 1.39·7-s − 0.662·9-s + 11-s + 6.05·13-s + 3.26·17-s − 19-s + 2.13·21-s − 0.528·23-s + 5.59·27-s + 1.60·29-s − 3.79·31-s − 1.52·33-s − 8.92·37-s − 9.26·39-s + 1.86·41-s − 0.866·43-s − 1.19·47-s − 5.05·49-s − 4.98·51-s + 10.7·53-s + 1.52·57-s − 13.1·59-s − 3.12·61-s + 0.924·63-s + 2.26·67-s + 0.808·69-s + ⋯ |
| L(s) = 1 | − 0.882·3-s − 0.527·7-s − 0.220·9-s + 0.301·11-s + 1.68·13-s + 0.791·17-s − 0.229·19-s + 0.465·21-s − 0.110·23-s + 1.07·27-s + 0.297·29-s − 0.680·31-s − 0.266·33-s − 1.46·37-s − 1.48·39-s + 0.291·41-s − 0.132·43-s − 0.173·47-s − 0.721·49-s − 0.698·51-s + 1.47·53-s + 0.202·57-s − 1.71·59-s − 0.400·61-s + 0.116·63-s + 0.276·67-s + 0.0973·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.196018329\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.196018329\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 1.52T + 3T^{2} \) |
| 7 | \( 1 + 1.39T + 7T^{2} \) |
| 13 | \( 1 - 6.05T + 13T^{2} \) |
| 17 | \( 1 - 3.26T + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 + 0.528T + 23T^{2} \) |
| 29 | \( 1 - 1.60T + 29T^{2} \) |
| 31 | \( 1 + 3.79T + 31T^{2} \) |
| 37 | \( 1 + 8.92T + 37T^{2} \) |
| 41 | \( 1 - 1.86T + 41T^{2} \) |
| 43 | \( 1 + 0.866T + 43T^{2} \) |
| 47 | \( 1 + 1.19T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 + 13.1T + 59T^{2} \) |
| 61 | \( 1 + 3.12T + 61T^{2} \) |
| 67 | \( 1 - 2.26T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 - 2.58T + 79T^{2} \) |
| 83 | \( 1 + 0.128T + 83T^{2} \) |
| 89 | \( 1 - 4.12T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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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
−8.525055985568093778870910447006, −7.55595351187053642979851721953, −6.66066647243147141247341440625, −6.11527136466225808308451137764, −5.61982520264766042764915625303, −4.74860295726542995863022934361, −3.69025954952521827178284207298, −3.15895914236259248250961976048, −1.72193191203450511306766954759, −0.65775892690977274096725515561,
0.65775892690977274096725515561, 1.72193191203450511306766954759, 3.15895914236259248250961976048, 3.69025954952521827178284207298, 4.74860295726542995863022934361, 5.61982520264766042764915625303, 6.11527136466225808308451137764, 6.66066647243147141247341440625, 7.55595351187053642979851721953, 8.525055985568093778870910447006
