| L(s) = 1 | + 2-s + 1.22·3-s + 4-s − 3.15·5-s + 1.22·6-s − 1.76·7-s + 8-s − 1.50·9-s − 3.15·10-s + 1.53·11-s + 1.22·12-s + 3.52·13-s − 1.76·14-s − 3.85·15-s + 16-s − 0.0414·17-s − 1.50·18-s + 0.885·19-s − 3.15·20-s − 2.15·21-s + 1.53·22-s − 23-s + 1.22·24-s + 4.96·25-s + 3.52·26-s − 5.50·27-s − 1.76·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.705·3-s + 0.5·4-s − 1.41·5-s + 0.499·6-s − 0.666·7-s + 0.353·8-s − 0.501·9-s − 0.998·10-s + 0.462·11-s + 0.352·12-s + 0.977·13-s − 0.471·14-s − 0.996·15-s + 0.250·16-s − 0.0100·17-s − 0.354·18-s + 0.203·19-s − 0.705·20-s − 0.470·21-s + 0.327·22-s − 0.208·23-s + 0.249·24-s + 0.992·25-s + 0.691·26-s − 1.05·27-s − 0.333·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.641702845\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.641702845\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 - 1.22T + 3T^{2} \) |
| 5 | \( 1 + 3.15T + 5T^{2} \) |
| 7 | \( 1 + 1.76T + 7T^{2} \) |
| 11 | \( 1 - 1.53T + 11T^{2} \) |
| 13 | \( 1 - 3.52T + 13T^{2} \) |
| 17 | \( 1 + 0.0414T + 17T^{2} \) |
| 19 | \( 1 - 0.885T + 19T^{2} \) |
| 29 | \( 1 + 2.60T + 29T^{2} \) |
| 31 | \( 1 - 1.22T + 31T^{2} \) |
| 37 | \( 1 - 5.19T + 37T^{2} \) |
| 41 | \( 1 - 4.97T + 41T^{2} \) |
| 43 | \( 1 - 1.11T + 43T^{2} \) |
| 47 | \( 1 + 8.28T + 47T^{2} \) |
| 53 | \( 1 - 1.16T + 53T^{2} \) |
| 59 | \( 1 - 7.31T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 - 14.3T + 67T^{2} \) |
| 71 | \( 1 + 2.70T + 71T^{2} \) |
| 73 | \( 1 + 0.781T + 73T^{2} \) |
| 79 | \( 1 - 15.7T + 79T^{2} \) |
| 83 | \( 1 + 2.75T + 83T^{2} \) |
| 89 | \( 1 - 0.259T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−8.045579130052252931314233971451, −7.45708982325117684416513685612, −6.59748919567113833326568005348, −6.04020114349647264839970757417, −5.09274823942158508798244239712, −4.07901337387250856590860112560, −3.65001647804972458393088482634, −3.15221860423354042340077201989, −2.16983865578333987133843551463, −0.72466211906181579154863157802,
0.72466211906181579154863157802, 2.16983865578333987133843551463, 3.15221860423354042340077201989, 3.65001647804972458393088482634, 4.07901337387250856590860112560, 5.09274823942158508798244239712, 6.04020114349647264839970757417, 6.59748919567113833326568005348, 7.45708982325117684416513685612, 8.045579130052252931314233971451
