| L(s) = 1 | + 2-s + 2.38·3-s + 4-s + 1.33·5-s + 2.38·6-s + 1.97·7-s + 8-s + 2.67·9-s + 1.33·10-s − 2.08·11-s + 2.38·12-s − 2.75·13-s + 1.97·14-s + 3.17·15-s + 16-s + 6.31·17-s + 2.67·18-s − 3.77·19-s + 1.33·20-s + 4.69·21-s − 2.08·22-s + 5.95·23-s + 2.38·24-s − 3.22·25-s − 2.75·26-s − 0.764·27-s + 1.97·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.37·3-s + 0.5·4-s + 0.595·5-s + 0.972·6-s + 0.744·7-s + 0.353·8-s + 0.893·9-s + 0.421·10-s − 0.627·11-s + 0.687·12-s − 0.764·13-s + 0.526·14-s + 0.819·15-s + 0.250·16-s + 1.53·17-s + 0.631·18-s − 0.866·19-s + 0.297·20-s + 1.02·21-s − 0.443·22-s + 1.24·23-s + 0.486·24-s − 0.645·25-s − 0.540·26-s − 0.147·27-s + 0.372·28-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
| \(L(1)\) |
\(\approx\) |
\(5.990989556\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.990989556\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.38T + 3T^{2} \) |
| 5 | \( 1 - 1.33T + 5T^{2} \) |
| 7 | \( 1 - 1.97T + 7T^{2} \) |
| 11 | \( 1 + 2.08T + 11T^{2} \) |
| 13 | \( 1 + 2.75T + 13T^{2} \) |
| 17 | \( 1 - 6.31T + 17T^{2} \) |
| 19 | \( 1 + 3.77T + 19T^{2} \) |
| 23 | \( 1 - 5.95T + 23T^{2} \) |
| 29 | \( 1 - 5.59T + 29T^{2} \) |
| 31 | \( 1 - 0.145T + 31T^{2} \) |
| 37 | \( 1 - 1.45T + 37T^{2} \) |
| 41 | \( 1 - 5.95T + 41T^{2} \) |
| 43 | \( 1 - 0.273T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 + 0.0332T + 53T^{2} \) |
| 59 | \( 1 + 3.86T + 59T^{2} \) |
| 61 | \( 1 + 6.92T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 - 13.6T + 73T^{2} \) |
| 79 | \( 1 + 4.54T + 79T^{2} \) |
| 83 | \( 1 - 11.6T + 83T^{2} \) |
| 89 | \( 1 + 2.26T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−8.224427222286192532977017700574, −7.81562285372186993617703052554, −7.17494305564465909984796557692, −6.11000905414156792337244287678, −5.32186456161862307705914017088, −4.66316857399216328824105500117, −3.74329704949606221667858679966, −2.78842152734951682847679415539, −2.38086307671210753586627932039, −1.33524420850790616560979860976,
1.33524420850790616560979860976, 2.38086307671210753586627932039, 2.78842152734951682847679415539, 3.74329704949606221667858679966, 4.66316857399216328824105500117, 5.32186456161862307705914017088, 6.11000905414156792337244287678, 7.17494305564465909984796557692, 7.81562285372186993617703052554, 8.224427222286192532977017700574
