| L(s) = 1 | + 2-s + 2.14·3-s + 4-s + 3.16·5-s + 2.14·6-s + 2.76·7-s + 8-s + 1.59·9-s + 3.16·10-s − 4.01·11-s + 2.14·12-s + 5.37·13-s + 2.76·14-s + 6.79·15-s + 16-s + 2.09·17-s + 1.59·18-s − 19-s + 3.16·20-s + 5.93·21-s − 4.01·22-s − 4.42·23-s + 2.14·24-s + 5.04·25-s + 5.37·26-s − 3.00·27-s + 2.76·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.23·3-s + 0.5·4-s + 1.41·5-s + 0.875·6-s + 1.04·7-s + 0.353·8-s + 0.532·9-s + 1.00·10-s − 1.21·11-s + 0.619·12-s + 1.49·13-s + 0.740·14-s + 1.75·15-s + 0.250·16-s + 0.508·17-s + 0.376·18-s − 0.229·19-s + 0.708·20-s + 1.29·21-s − 0.856·22-s − 0.922·23-s + 0.437·24-s + 1.00·25-s + 1.05·26-s − 0.578·27-s + 0.523·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.814900380\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.814900380\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 - 2.14T + 3T^{2} \) |
| 5 | \( 1 - 3.16T + 5T^{2} \) |
| 7 | \( 1 - 2.76T + 7T^{2} \) |
| 11 | \( 1 + 4.01T + 11T^{2} \) |
| 13 | \( 1 - 5.37T + 13T^{2} \) |
| 17 | \( 1 - 2.09T + 17T^{2} \) |
| 23 | \( 1 + 4.42T + 23T^{2} \) |
| 29 | \( 1 + 8.75T + 29T^{2} \) |
| 31 | \( 1 + 6.73T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 + 0.355T + 41T^{2} \) |
| 43 | \( 1 - 8.11T + 43T^{2} \) |
| 47 | \( 1 - 5.05T + 47T^{2} \) |
| 53 | \( 1 - 0.926T + 53T^{2} \) |
| 59 | \( 1 - 8.02T + 59T^{2} \) |
| 61 | \( 1 - 5.90T + 61T^{2} \) |
| 67 | \( 1 - 3.57T + 67T^{2} \) |
| 71 | \( 1 + 2.04T + 71T^{2} \) |
| 73 | \( 1 - 4.11T + 73T^{2} \) |
| 79 | \( 1 + 0.388T + 79T^{2} \) |
| 83 | \( 1 - 10.9T + 83T^{2} \) |
| 89 | \( 1 + 13.9T + 89T^{2} \) |
| 97 | \( 1 - 2.38T + 97T^{2} \) |
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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
−7.86054457997787750377716426212, −7.34955097606568523520841869424, −6.11158621809925168518935617757, −5.71286485609170842883074898234, −5.20627789658811753299623370444, −4.09097053523987678333707765660, −3.54641046137987061788906722130, −2.43161529887406397220148657816, −2.17000198096109227571556797315, −1.31967260134287636779770289313,
1.31967260134287636779770289313, 2.17000198096109227571556797315, 2.43161529887406397220148657816, 3.54641046137987061788906722130, 4.09097053523987678333707765660, 5.20627789658811753299623370444, 5.71286485609170842883074898234, 6.11158621809925168518935617757, 7.34955097606568523520841869424, 7.86054457997787750377716426212
