| L(s) = 1 | + 2-s + 2.19·3-s + 4-s + 3.12·5-s + 2.19·6-s + 2.17·7-s + 8-s + 1.82·9-s + 3.12·10-s + 3.12·11-s + 2.19·12-s + 2.17·14-s + 6.87·15-s + 16-s − 7.10·17-s + 1.82·18-s − 19-s + 3.12·20-s + 4.78·21-s + 3.12·22-s − 0.644·23-s + 2.19·24-s + 4.77·25-s − 2.57·27-s + 2.17·28-s + 0.426·29-s + 6.87·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.26·3-s + 0.5·4-s + 1.39·5-s + 0.897·6-s + 0.823·7-s + 0.353·8-s + 0.609·9-s + 0.988·10-s + 0.942·11-s + 0.634·12-s + 0.582·14-s + 1.77·15-s + 0.250·16-s − 1.72·17-s + 0.431·18-s − 0.229·19-s + 0.699·20-s + 1.04·21-s + 0.666·22-s − 0.134·23-s + 0.448·24-s + 0.955·25-s − 0.494·27-s + 0.411·28-s + 0.0791·29-s + 1.25·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.509225094\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.509225094\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 13 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 2.19T + 3T^{2} \) |
| 5 | \( 1 - 3.12T + 5T^{2} \) |
| 7 | \( 1 - 2.17T + 7T^{2} \) |
| 11 | \( 1 - 3.12T + 11T^{2} \) |
| 17 | \( 1 + 7.10T + 17T^{2} \) |
| 23 | \( 1 + 0.644T + 23T^{2} \) |
| 29 | \( 1 - 0.426T + 29T^{2} \) |
| 31 | \( 1 + 4.85T + 31T^{2} \) |
| 37 | \( 1 - 7.52T + 37T^{2} \) |
| 41 | \( 1 + 0.117T + 41T^{2} \) |
| 43 | \( 1 - 2.42T + 43T^{2} \) |
| 47 | \( 1 - 6.12T + 47T^{2} \) |
| 53 | \( 1 - 4.11T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 - 8.65T + 61T^{2} \) |
| 67 | \( 1 - 1.75T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 - 1.43T + 89T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
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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
−8.139908479588273814884799004203, −7.23627369919223893502155189951, −6.52766397678108108690334003126, −5.92910711969331225063215585962, −5.09156109778315568910720647608, −4.29414956946876486818757811290, −3.67652333892498378981601400851, −2.45624618924752629617932711693, −2.21824389704580148469048776718, −1.38579017778443513079370701857,
1.38579017778443513079370701857, 2.21824389704580148469048776718, 2.45624618924752629617932711693, 3.67652333892498378981601400851, 4.29414956946876486818757811290, 5.09156109778315568910720647608, 5.92910711969331225063215585962, 6.52766397678108108690334003126, 7.23627369919223893502155189951, 8.139908479588273814884799004203
