| L(s) = 1 | + 2-s − 1.93·3-s + 4-s − 3.29·5-s − 1.93·6-s − 4.13·7-s + 8-s + 0.763·9-s − 3.29·10-s − 1.37·11-s − 1.93·12-s − 4.99·13-s − 4.13·14-s + 6.38·15-s + 16-s − 17-s + 0.763·18-s + 7.53·19-s − 3.29·20-s + 8.01·21-s − 1.37·22-s − 8.63·23-s − 1.93·24-s + 5.83·25-s − 4.99·26-s + 4.33·27-s − 4.13·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.12·3-s + 0.5·4-s − 1.47·5-s − 0.792·6-s − 1.56·7-s + 0.353·8-s + 0.254·9-s − 1.04·10-s − 0.414·11-s − 0.560·12-s − 1.38·13-s − 1.10·14-s + 1.64·15-s + 0.250·16-s − 0.242·17-s + 0.179·18-s + 1.72·19-s − 0.736·20-s + 1.74·21-s − 0.293·22-s − 1.80·23-s − 0.396·24-s + 1.16·25-s − 0.979·26-s + 0.834·27-s − 0.780·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3609445097\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3609445097\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 + 1.93T + 3T^{2} \) |
| 5 | \( 1 + 3.29T + 5T^{2} \) |
| 7 | \( 1 + 4.13T + 7T^{2} \) |
| 11 | \( 1 + 1.37T + 11T^{2} \) |
| 13 | \( 1 + 4.99T + 13T^{2} \) |
| 19 | \( 1 - 7.53T + 19T^{2} \) |
| 23 | \( 1 + 8.63T + 23T^{2} \) |
| 29 | \( 1 + 8.59T + 29T^{2} \) |
| 31 | \( 1 + 7.14T + 31T^{2} \) |
| 37 | \( 1 - 7.68T + 37T^{2} \) |
| 41 | \( 1 - 5.60T + 41T^{2} \) |
| 43 | \( 1 + 9.03T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 + 4.82T + 53T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 - 5.22T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 - 0.929T + 73T^{2} \) |
| 79 | \( 1 - 0.584T + 79T^{2} \) |
| 83 | \( 1 + 8.14T + 83T^{2} \) |
| 89 | \( 1 + 4.62T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−9.479490635774149381818078871730, −7.993859767590418151723300118481, −7.35887567338211076523325569043, −6.79893328415039057554672918119, −5.75131108779413526081509065771, −5.30944419556398487455076040700, −4.17224551702785323070164515104, −3.54012385427196517578079500745, −2.57322044770570809840161642654, −0.35182137611074110413957339569,
0.35182137611074110413957339569, 2.57322044770570809840161642654, 3.54012385427196517578079500745, 4.17224551702785323070164515104, 5.30944419556398487455076040700, 5.75131108779413526081509065771, 6.79893328415039057554672918119, 7.35887567338211076523325569043, 7.993859767590418151723300118481, 9.479490635774149381818078871730
