| L(s) = 1 | − 2-s − 2.53·3-s + 4-s − 2.12·5-s + 2.53·6-s − 1.20·7-s − 8-s + 3.41·9-s + 2.12·10-s − 3.06·11-s − 2.53·12-s + 2.94·13-s + 1.20·14-s + 5.39·15-s + 16-s − 0.513·17-s − 3.41·18-s + 4.03·19-s − 2.12·20-s + 3.04·21-s + 3.06·22-s + 6.16·23-s + 2.53·24-s − 0.466·25-s − 2.94·26-s − 1.05·27-s − 1.20·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.46·3-s + 0.5·4-s − 0.952·5-s + 1.03·6-s − 0.455·7-s − 0.353·8-s + 1.13·9-s + 0.673·10-s − 0.924·11-s − 0.731·12-s + 0.815·13-s + 0.321·14-s + 1.39·15-s + 0.250·16-s − 0.124·17-s − 0.805·18-s + 0.925·19-s − 0.476·20-s + 0.665·21-s + 0.653·22-s + 1.28·23-s + 0.517·24-s − 0.0932·25-s − 0.576·26-s − 0.203·27-s − 0.227·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4782816199\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4782816199\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 4021 | \( 1+O(T) \) |
| good | 3 | \( 1 + 2.53T + 3T^{2} \) |
| 5 | \( 1 + 2.12T + 5T^{2} \) |
| 7 | \( 1 + 1.20T + 7T^{2} \) |
| 11 | \( 1 + 3.06T + 11T^{2} \) |
| 13 | \( 1 - 2.94T + 13T^{2} \) |
| 17 | \( 1 + 0.513T + 17T^{2} \) |
| 19 | \( 1 - 4.03T + 19T^{2} \) |
| 23 | \( 1 - 6.16T + 23T^{2} \) |
| 29 | \( 1 - 0.113T + 29T^{2} \) |
| 31 | \( 1 - 6.64T + 31T^{2} \) |
| 37 | \( 1 - 1.12T + 37T^{2} \) |
| 41 | \( 1 - 7.47T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 + 9.30T + 47T^{2} \) |
| 53 | \( 1 + 0.632T + 53T^{2} \) |
| 59 | \( 1 + 12.6T + 59T^{2} \) |
| 61 | \( 1 - 8.24T + 61T^{2} \) |
| 67 | \( 1 - 3.19T + 67T^{2} \) |
| 71 | \( 1 + 5.15T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 - 6.64T + 79T^{2} \) |
| 83 | \( 1 - 3.76T + 83T^{2} \) |
| 89 | \( 1 + 5.64T + 89T^{2} \) |
| 97 | \( 1 - 18.1T + 97T^{2} \) |
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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
−7.66583252592594966823063361725, −7.26850794329244787799640408914, −6.35138051324873826913330783127, −5.97938609408577630934551145726, −5.09768583356576027680487760621, −4.48783397850262018237446051863, −3.44230611014770201539739726261, −2.71214186356371678761109625589, −1.19719511996817725962087718534, −0.48163630574511201631146758957,
0.48163630574511201631146758957, 1.19719511996817725962087718534, 2.71214186356371678761109625589, 3.44230611014770201539739726261, 4.48783397850262018237446051863, 5.09768583356576027680487760621, 5.97938609408577630934551145726, 6.35138051324873826913330783127, 7.26850794329244787799640408914, 7.66583252592594966823063361725
