| L(s) = 1 | + 0.659·3-s + 0.997·5-s + 0.366·7-s − 2.56·9-s + 2.34·11-s + 1.58·13-s + 0.657·15-s + 0.164·17-s − 2.94·19-s + 0.241·21-s − 6.49·23-s − 4.00·25-s − 3.66·27-s − 7.89·29-s − 7.02·31-s + 1.54·33-s + 0.365·35-s − 5.39·37-s + 1.04·39-s − 4.46·41-s + 8.37·43-s − 2.56·45-s − 12.8·47-s − 6.86·49-s + 0.108·51-s + 12.7·53-s + 2.33·55-s + ⋯ |
| L(s) = 1 | + 0.380·3-s + 0.446·5-s + 0.138·7-s − 0.855·9-s + 0.706·11-s + 0.439·13-s + 0.169·15-s + 0.0398·17-s − 0.674·19-s + 0.0527·21-s − 1.35·23-s − 0.800·25-s − 0.705·27-s − 1.46·29-s − 1.26·31-s + 0.268·33-s + 0.0618·35-s − 0.886·37-s + 0.167·39-s − 0.697·41-s + 1.27·43-s − 0.381·45-s − 1.87·47-s − 0.980·49-s + 0.0151·51-s + 1.74·53-s + 0.315·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 - 0.659T + 3T^{2} \) |
| 5 | \( 1 - 0.997T + 5T^{2} \) |
| 7 | \( 1 - 0.366T + 7T^{2} \) |
| 11 | \( 1 - 2.34T + 11T^{2} \) |
| 13 | \( 1 - 1.58T + 13T^{2} \) |
| 17 | \( 1 - 0.164T + 17T^{2} \) |
| 19 | \( 1 + 2.94T + 19T^{2} \) |
| 23 | \( 1 + 6.49T + 23T^{2} \) |
| 29 | \( 1 + 7.89T + 29T^{2} \) |
| 31 | \( 1 + 7.02T + 31T^{2} \) |
| 37 | \( 1 + 5.39T + 37T^{2} \) |
| 41 | \( 1 + 4.46T + 41T^{2} \) |
| 43 | \( 1 - 8.37T + 43T^{2} \) |
| 47 | \( 1 + 12.8T + 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 - 9.64T + 59T^{2} \) |
| 61 | \( 1 - 9.98T + 61T^{2} \) |
| 67 | \( 1 + 6.58T + 67T^{2} \) |
| 71 | \( 1 - 9.53T + 71T^{2} \) |
| 73 | \( 1 - 2.32T + 73T^{2} \) |
| 79 | \( 1 - 2.51T + 79T^{2} \) |
| 83 | \( 1 + 2.90T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 - 0.261T + 97T^{2} \) |
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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
−8.273857070336396683778991148174, −7.40445248287318123008742946267, −6.53469900946329860245019507821, −5.81363417538508944541176617245, −5.29867684683660450989959598583, −3.94403630444740715866537401894, −3.60832880254861696259748314352, −2.28989659688338741813505677511, −1.70098313496692734160949122864, 0,
1.70098313496692734160949122864, 2.28989659688338741813505677511, 3.60832880254861696259748314352, 3.94403630444740715866537401894, 5.29867684683660450989959598583, 5.81363417538508944541176617245, 6.53469900946329860245019507821, 7.40445248287318123008742946267, 8.273857070336396683778991148174
