L(s) = 1 | + 2-s − 2.90·3-s + 4-s − 5-s − 2.90·6-s + 4.43·7-s + 8-s + 5.42·9-s − 10-s + 4.85·11-s − 2.90·12-s + 1.74·13-s + 4.43·14-s + 2.90·15-s + 16-s + 0.344·17-s + 5.42·18-s + 2.92·19-s − 20-s − 12.8·21-s + 4.85·22-s + 8.04·23-s − 2.90·24-s + 25-s + 1.74·26-s − 7.04·27-s + 4.43·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.67·3-s + 0.5·4-s − 0.447·5-s − 1.18·6-s + 1.67·7-s + 0.353·8-s + 1.80·9-s − 0.316·10-s + 1.46·11-s − 0.837·12-s + 0.483·13-s + 1.18·14-s + 0.749·15-s + 0.250·16-s + 0.0835·17-s + 1.27·18-s + 0.670·19-s − 0.223·20-s − 2.81·21-s + 1.03·22-s + 1.67·23-s − 0.592·24-s + 0.200·25-s + 0.341·26-s − 1.35·27-s + 0.838·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.488453649\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.488453649\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 401 | \( 1 + T \) |
good | 3 | \( 1 + 2.90T + 3T^{2} \) |
| 7 | \( 1 - 4.43T + 7T^{2} \) |
| 11 | \( 1 - 4.85T + 11T^{2} \) |
| 13 | \( 1 - 1.74T + 13T^{2} \) |
| 17 | \( 1 - 0.344T + 17T^{2} \) |
| 19 | \( 1 - 2.92T + 19T^{2} \) |
| 23 | \( 1 - 8.04T + 23T^{2} \) |
| 29 | \( 1 + 1.84T + 29T^{2} \) |
| 31 | \( 1 - 8.68T + 31T^{2} \) |
| 37 | \( 1 - 8.18T + 37T^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 + 12.7T + 43T^{2} \) |
| 47 | \( 1 - 4.01T + 47T^{2} \) |
| 53 | \( 1 + 3.27T + 53T^{2} \) |
| 59 | \( 1 + 5.69T + 59T^{2} \) |
| 61 | \( 1 + 9.09T + 61T^{2} \) |
| 67 | \( 1 - 2.58T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 + 1.11T + 73T^{2} \) |
| 79 | \( 1 - 2.17T + 79T^{2} \) |
| 83 | \( 1 - 0.0749T + 83T^{2} \) |
| 89 | \( 1 - 10.0T + 89T^{2} \) |
| 97 | \( 1 + 1.03T + 97T^{2} \) |
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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
−8.273184680395369456751532745907, −7.45528158346381375542046240739, −6.72891882908883511467472172380, −6.19334029250447238566859974623, −5.27878651514199955947368561479, −4.76367860822722423159644948655, −4.29627630135627302891955786222, −3.23290991826367710266939671162, −1.52480233062643894164250332797, −1.05179097630922775349409706261,
1.05179097630922775349409706261, 1.52480233062643894164250332797, 3.23290991826367710266939671162, 4.29627630135627302891955786222, 4.76367860822722423159644948655, 5.27878651514199955947368561479, 6.19334029250447238566859974623, 6.72891882908883511467472172380, 7.45528158346381375542046240739, 8.273184680395369456751532745907