| L(s) = 1 | + 1.69·2-s + 1.71·3-s + 0.876·4-s − 0.385·5-s + 2.90·6-s + 3.49·7-s − 1.90·8-s − 0.0652·9-s − 0.653·10-s + 4.33·11-s + 1.50·12-s − 1.27·13-s + 5.92·14-s − 0.660·15-s − 4.98·16-s + 0.562·17-s − 0.110·18-s + 7.63·19-s − 0.337·20-s + 5.98·21-s + 7.35·22-s + 7.41·23-s − 3.26·24-s − 4.85·25-s − 2.16·26-s − 5.25·27-s + 3.05·28-s + ⋯ |
| L(s) = 1 | + 1.19·2-s + 0.989·3-s + 0.438·4-s − 0.172·5-s + 1.18·6-s + 1.31·7-s − 0.673·8-s − 0.0217·9-s − 0.206·10-s + 1.30·11-s + 0.433·12-s − 0.354·13-s + 1.58·14-s − 0.170·15-s − 1.24·16-s + 0.136·17-s − 0.0260·18-s + 1.75·19-s − 0.0755·20-s + 1.30·21-s + 1.56·22-s + 1.54·23-s − 0.666·24-s − 0.970·25-s − 0.425·26-s − 1.01·27-s + 0.578·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.795114853\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.795114853\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 - 1.69T + 2T^{2} \) |
| 3 | \( 1 - 1.71T + 3T^{2} \) |
| 5 | \( 1 + 0.385T + 5T^{2} \) |
| 7 | \( 1 - 3.49T + 7T^{2} \) |
| 11 | \( 1 - 4.33T + 11T^{2} \) |
| 13 | \( 1 + 1.27T + 13T^{2} \) |
| 17 | \( 1 - 0.562T + 17T^{2} \) |
| 19 | \( 1 - 7.63T + 19T^{2} \) |
| 23 | \( 1 - 7.41T + 23T^{2} \) |
| 29 | \( 1 + 3.41T + 29T^{2} \) |
| 31 | \( 1 - 0.978T + 31T^{2} \) |
| 37 | \( 1 + 4.39T + 37T^{2} \) |
| 41 | \( 1 - 4.85T + 41T^{2} \) |
| 47 | \( 1 + 4.37T + 47T^{2} \) |
| 53 | \( 1 - 9.67T + 53T^{2} \) |
| 59 | \( 1 + 4.68T + 59T^{2} \) |
| 61 | \( 1 - 5.41T + 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 + 7.63T + 71T^{2} \) |
| 73 | \( 1 + 2.07T + 73T^{2} \) |
| 79 | \( 1 + 8.28T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 + 2.72T + 89T^{2} \) |
| 97 | \( 1 + 0.490T + 97T^{2} \) |
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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
−9.135973941122783307970361552025, −8.505706561347388377106173788072, −7.62479729611072700156851404099, −6.93445913042613312093726316308, −5.66283148352959655086365523671, −5.11903253082324976372266919091, −4.15670261453655936747454161948, −3.50367860740217748132359135389, −2.62821219999899869312130035951, −1.40318375804221736789432919402,
1.40318375804221736789432919402, 2.62821219999899869312130035951, 3.50367860740217748132359135389, 4.15670261453655936747454161948, 5.11903253082324976372266919091, 5.66283148352959655086365523671, 6.93445913042613312093726316308, 7.62479729611072700156851404099, 8.505706561347388377106173788072, 9.135973941122783307970361552025
