| L(s) = 1 | + 2-s + 0.0387·3-s + 4-s + 2.61·5-s + 0.0387·6-s + 8-s − 2.99·9-s + 2.61·10-s + 3.07·11-s + 0.0387·12-s − 2.08·13-s + 0.101·15-s + 16-s − 0.761·17-s − 2.99·18-s + 7.09·19-s + 2.61·20-s + 3.07·22-s + 5.31·23-s + 0.0387·24-s + 1.81·25-s − 2.08·26-s − 0.232·27-s + 2.31·29-s + 0.101·30-s + 0.644·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.0223·3-s + 0.5·4-s + 1.16·5-s + 0.0158·6-s + 0.353·8-s − 0.999·9-s + 0.825·10-s + 0.926·11-s + 0.0111·12-s − 0.579·13-s + 0.0261·15-s + 0.250·16-s − 0.184·17-s − 0.706·18-s + 1.62·19-s + 0.583·20-s + 0.654·22-s + 1.10·23-s + 0.00791·24-s + 0.362·25-s − 0.409·26-s − 0.0447·27-s + 0.429·29-s + 0.0184·30-s + 0.115·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.960954093\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.960954093\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 - 0.0387T + 3T^{2} \) |
| 5 | \( 1 - 2.61T + 5T^{2} \) |
| 11 | \( 1 - 3.07T + 11T^{2} \) |
| 13 | \( 1 + 2.08T + 13T^{2} \) |
| 17 | \( 1 + 0.761T + 17T^{2} \) |
| 19 | \( 1 - 7.09T + 19T^{2} \) |
| 23 | \( 1 - 5.31T + 23T^{2} \) |
| 29 | \( 1 - 2.31T + 29T^{2} \) |
| 31 | \( 1 - 0.644T + 31T^{2} \) |
| 37 | \( 1 + 9.45T + 37T^{2} \) |
| 43 | \( 1 - 4.79T + 43T^{2} \) |
| 47 | \( 1 - 4.61T + 47T^{2} \) |
| 53 | \( 1 + 6.27T + 53T^{2} \) |
| 59 | \( 1 - 14.3T + 59T^{2} \) |
| 61 | \( 1 - 3.84T + 61T^{2} \) |
| 67 | \( 1 - 8.37T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 - 3.35T + 73T^{2} \) |
| 79 | \( 1 + 2.24T + 79T^{2} \) |
| 83 | \( 1 - 7.80T + 83T^{2} \) |
| 89 | \( 1 + 5.93T + 89T^{2} \) |
| 97 | \( 1 - 16.7T + 97T^{2} \) |
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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
−8.595817119403047648705714074262, −7.48019567997483901866871859356, −6.81226359579042254161072019159, −6.11541530859087327122375767614, −5.36451569513370773114117748573, −4.99422779604706760820450661622, −3.74191863720047945034800232587, −2.95604806192576808530986240153, −2.17727802723836070404764206846, −1.08425862357158940386181237072,
1.08425862357158940386181237072, 2.17727802723836070404764206846, 2.95604806192576808530986240153, 3.74191863720047945034800232587, 4.99422779604706760820450661622, 5.36451569513370773114117748573, 6.11541530859087327122375767614, 6.81226359579042254161072019159, 7.48019567997483901866871859356, 8.595817119403047648705714074262
