L(s) = 1 | + 2-s + 0.995·3-s + 4-s + 3.65·5-s + 0.995·6-s + 3.82·7-s + 8-s − 2.00·9-s + 3.65·10-s + 1.37·11-s + 0.995·12-s + 0.314·13-s + 3.82·14-s + 3.64·15-s + 16-s − 0.627·17-s − 2.00·18-s − 5.51·19-s + 3.65·20-s + 3.80·21-s + 1.37·22-s + 3.26·23-s + 0.995·24-s + 8.36·25-s + 0.314·26-s − 4.98·27-s + 3.82·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.574·3-s + 0.5·4-s + 1.63·5-s + 0.406·6-s + 1.44·7-s + 0.353·8-s − 0.669·9-s + 1.15·10-s + 0.415·11-s + 0.287·12-s + 0.0873·13-s + 1.02·14-s + 0.939·15-s + 0.250·16-s − 0.152·17-s − 0.473·18-s − 1.26·19-s + 0.817·20-s + 0.831·21-s + 0.294·22-s + 0.679·23-s + 0.203·24-s + 1.67·25-s + 0.0617·26-s − 0.959·27-s + 0.722·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.789021945\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.789021945\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 2011 | \( 1 + T \) |
good | 3 | \( 1 - 0.995T + 3T^{2} \) |
| 5 | \( 1 - 3.65T + 5T^{2} \) |
| 7 | \( 1 - 3.82T + 7T^{2} \) |
| 11 | \( 1 - 1.37T + 11T^{2} \) |
| 13 | \( 1 - 0.314T + 13T^{2} \) |
| 17 | \( 1 + 0.627T + 17T^{2} \) |
| 19 | \( 1 + 5.51T + 19T^{2} \) |
| 23 | \( 1 - 3.26T + 23T^{2} \) |
| 29 | \( 1 + 4.51T + 29T^{2} \) |
| 31 | \( 1 + 0.743T + 31T^{2} \) |
| 37 | \( 1 - 3.92T + 37T^{2} \) |
| 41 | \( 1 + 4.88T + 41T^{2} \) |
| 43 | \( 1 - 11.9T + 43T^{2} \) |
| 47 | \( 1 + 8.55T + 47T^{2} \) |
| 53 | \( 1 - 2.25T + 53T^{2} \) |
| 59 | \( 1 - 2.36T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 + 7.35T + 67T^{2} \) |
| 71 | \( 1 + 1.41T + 71T^{2} \) |
| 73 | \( 1 + 6.82T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 - 9.91T + 83T^{2} \) |
| 89 | \( 1 + 14.4T + 89T^{2} \) |
| 97 | \( 1 + 0.544T + 97T^{2} \) |
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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
−8.583131992952776853222011186563, −7.73905144793910925439232573621, −6.81935029865729828514231208922, −6.01396100538937174664356269488, −5.49707854075591242862574164387, −4.77032754234713325252449761088, −3.92408723769212498757255173282, −2.72361317563570322563256122432, −2.11846512190846563985216954299, −1.42402589193504963188866504784,
1.42402589193504963188866504784, 2.11846512190846563985216954299, 2.72361317563570322563256122432, 3.92408723769212498757255173282, 4.77032754234713325252449761088, 5.49707854075591242862574164387, 6.01396100538937174664356269488, 6.81935029865729828514231208922, 7.73905144793910925439232573621, 8.583131992952776853222011186563