| L(s) = 1 | − 2-s − 1.79·3-s + 4-s + 3.03·5-s + 1.79·6-s − 2.51·7-s − 8-s + 0.232·9-s − 3.03·10-s − 4.37·11-s − 1.79·12-s + 4.32·13-s + 2.51·14-s − 5.45·15-s + 16-s + 4.54·17-s − 0.232·18-s + 3.86·19-s + 3.03·20-s + 4.53·21-s + 4.37·22-s − 7.19·23-s + 1.79·24-s + 4.21·25-s − 4.32·26-s + 4.97·27-s − 2.51·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.03·3-s + 0.5·4-s + 1.35·5-s + 0.733·6-s − 0.952·7-s − 0.353·8-s + 0.0773·9-s − 0.959·10-s − 1.32·11-s − 0.518·12-s + 1.19·13-s + 0.673·14-s − 1.40·15-s + 0.250·16-s + 1.10·17-s − 0.0546·18-s + 0.885·19-s + 0.678·20-s + 0.988·21-s + 0.933·22-s − 1.50·23-s + 0.366·24-s + 0.842·25-s − 0.848·26-s + 0.957·27-s − 0.476·28-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
| \(L(1)\) |
\(\approx\) |
\(0.8398730288\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8398730288\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.79T + 3T^{2} \) |
| 5 | \( 1 - 3.03T + 5T^{2} \) |
| 7 | \( 1 + 2.51T + 7T^{2} \) |
| 11 | \( 1 + 4.37T + 11T^{2} \) |
| 13 | \( 1 - 4.32T + 13T^{2} \) |
| 17 | \( 1 - 4.54T + 17T^{2} \) |
| 19 | \( 1 - 3.86T + 19T^{2} \) |
| 23 | \( 1 + 7.19T + 23T^{2} \) |
| 29 | \( 1 + 7.60T + 29T^{2} \) |
| 31 | \( 1 - 5.99T + 31T^{2} \) |
| 37 | \( 1 + 8.12T + 37T^{2} \) |
| 41 | \( 1 - 3.02T + 41T^{2} \) |
| 43 | \( 1 - 3.21T + 43T^{2} \) |
| 47 | \( 1 + 4.44T + 47T^{2} \) |
| 53 | \( 1 - 2.61T + 53T^{2} \) |
| 59 | \( 1 + 2.50T + 59T^{2} \) |
| 61 | \( 1 - 10.7T + 61T^{2} \) |
| 67 | \( 1 - 5.35T + 67T^{2} \) |
| 71 | \( 1 + 2.33T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 - 6.84T + 79T^{2} \) |
| 83 | \( 1 + 9.19T + 83T^{2} \) |
| 89 | \( 1 - 0.403T + 89T^{2} \) |
| 97 | \( 1 + 5.55T + 97T^{2} \) |
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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
−8.450071849529742182656120645278, −7.77538817691977142417629511600, −6.77865032070652120217963047044, −6.10502870143282415962337847639, −5.65681032851272593042198904231, −5.24651836019032471811708791284, −3.63723486104426758926598626850, −2.78230061081533167104886591753, −1.75457289678458522255162102772, −0.61013461523849099709964403176,
0.61013461523849099709964403176, 1.75457289678458522255162102772, 2.78230061081533167104886591753, 3.63723486104426758926598626850, 5.24651836019032471811708791284, 5.65681032851272593042198904231, 6.10502870143282415962337847639, 6.77865032070652120217963047044, 7.77538817691977142417629511600, 8.450071849529742182656120645278
