| L(s) = 1 | + 0.173·2-s + 0.896·3-s − 1.96·4-s − 3.74·5-s + 0.155·6-s − 1.28·7-s − 0.688·8-s − 2.19·9-s − 0.650·10-s − 4.72·11-s − 1.76·12-s − 4.53·13-s − 0.223·14-s − 3.36·15-s + 3.82·16-s + 2.76·17-s − 0.380·18-s − 2.71·19-s + 7.38·20-s − 1.15·21-s − 0.818·22-s + 4.99·23-s − 0.617·24-s + 9.06·25-s − 0.786·26-s − 4.65·27-s + 2.53·28-s + ⋯ |
| L(s) = 1 | + 0.122·2-s + 0.517·3-s − 0.984·4-s − 1.67·5-s + 0.0634·6-s − 0.486·7-s − 0.243·8-s − 0.732·9-s − 0.205·10-s − 1.42·11-s − 0.509·12-s − 1.25·13-s − 0.0596·14-s − 0.867·15-s + 0.955·16-s + 0.669·17-s − 0.0897·18-s − 0.623·19-s + 1.65·20-s − 0.251·21-s − 0.174·22-s + 1.04·23-s − 0.125·24-s + 1.81·25-s − 0.154·26-s − 0.896·27-s + 0.479·28-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
| \(L(1)\) |
\(\approx\) |
\(0.3816149315\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3816149315\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.173T + 2T^{2} \) |
| 3 | \( 1 - 0.896T + 3T^{2} \) |
| 5 | \( 1 + 3.74T + 5T^{2} \) |
| 7 | \( 1 + 1.28T + 7T^{2} \) |
| 11 | \( 1 + 4.72T + 11T^{2} \) |
| 13 | \( 1 + 4.53T + 13T^{2} \) |
| 17 | \( 1 - 2.76T + 17T^{2} \) |
| 19 | \( 1 + 2.71T + 19T^{2} \) |
| 23 | \( 1 - 4.99T + 23T^{2} \) |
| 29 | \( 1 + 1.47T + 29T^{2} \) |
| 31 | \( 1 - 3.76T + 31T^{2} \) |
| 37 | \( 1 - 5.53T + 37T^{2} \) |
| 41 | \( 1 - 1.04T + 41T^{2} \) |
| 47 | \( 1 - 2.04T + 47T^{2} \) |
| 53 | \( 1 - 1.20T + 53T^{2} \) |
| 59 | \( 1 + 0.436T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 - 7.86T + 67T^{2} \) |
| 71 | \( 1 + 15.4T + 71T^{2} \) |
| 73 | \( 1 + 15.5T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 - 0.437T + 83T^{2} \) |
| 89 | \( 1 - 5.41T + 89T^{2} \) |
| 97 | \( 1 - 6.46T + 97T^{2} \) |
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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
−9.061425321245463371064492337624, −8.374384850530805693472737590560, −7.77868666293300810955285569423, −7.30621377338339339252524241635, −5.86078002842151270573626564041, −4.92331199762772162434955042566, −4.33096187477065587862898689460, −3.21067868758509068877004283382, −2.81225033874111744754849783171, −0.37750543834778625090246049285,
0.37750543834778625090246049285, 2.81225033874111744754849783171, 3.21067868758509068877004283382, 4.33096187477065587862898689460, 4.92331199762772162434955042566, 5.86078002842151270573626564041, 7.30621377338339339252524241635, 7.77868666293300810955285569423, 8.374384850530805693472737590560, 9.061425321245463371064492337624
