| L(s) = 1 | − 2-s + 3.19·3-s + 4-s + 2.51·5-s − 3.19·6-s + 2.15·7-s − 8-s + 7.23·9-s − 2.51·10-s − 2.07·11-s + 3.19·12-s − 4.87·13-s − 2.15·14-s + 8.04·15-s + 16-s + 2.20·17-s − 7.23·18-s − 0.535·19-s + 2.51·20-s + 6.89·21-s + 2.07·22-s + 3.57·23-s − 3.19·24-s + 1.32·25-s + 4.87·26-s + 13.5·27-s + 2.15·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.84·3-s + 0.5·4-s + 1.12·5-s − 1.30·6-s + 0.814·7-s − 0.353·8-s + 2.41·9-s − 0.795·10-s − 0.625·11-s + 0.923·12-s − 1.35·13-s − 0.576·14-s + 2.07·15-s + 0.250·16-s + 0.535·17-s − 1.70·18-s − 0.122·19-s + 0.562·20-s + 1.50·21-s + 0.442·22-s + 0.746·23-s − 0.653·24-s + 0.265·25-s + 0.955·26-s + 2.60·27-s + 0.407·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.754408039\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.754408039\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2003 | \( 1 - T \) |
| good | 3 | \( 1 - 3.19T + 3T^{2} \) |
| 5 | \( 1 - 2.51T + 5T^{2} \) |
| 7 | \( 1 - 2.15T + 7T^{2} \) |
| 11 | \( 1 + 2.07T + 11T^{2} \) |
| 13 | \( 1 + 4.87T + 13T^{2} \) |
| 17 | \( 1 - 2.20T + 17T^{2} \) |
| 19 | \( 1 + 0.535T + 19T^{2} \) |
| 23 | \( 1 - 3.57T + 23T^{2} \) |
| 29 | \( 1 + 1.53T + 29T^{2} \) |
| 31 | \( 1 - 8.32T + 31T^{2} \) |
| 37 | \( 1 + 0.777T + 37T^{2} \) |
| 41 | \( 1 + 9.49T + 41T^{2} \) |
| 43 | \( 1 - 5.12T + 43T^{2} \) |
| 47 | \( 1 + 1.51T + 47T^{2} \) |
| 53 | \( 1 - 4.58T + 53T^{2} \) |
| 59 | \( 1 - 8.10T + 59T^{2} \) |
| 61 | \( 1 + 6.58T + 61T^{2} \) |
| 67 | \( 1 - 8.53T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 - 4.48T + 73T^{2} \) |
| 79 | \( 1 - 5.13T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 - 4.60T + 89T^{2} \) |
| 97 | \( 1 + 0.536T + 97T^{2} \) |
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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
−8.310709542379368773827746750368, −8.036484347716587976110533538713, −7.28034036221079194006370691208, −6.61487682809030095055817242167, −5.33295213934120228720826206170, −4.72173984764811909200463972071, −3.48355439196781874323725997784, −2.50203751714958064122409856717, −2.21815726173168531504182273995, −1.24227869265513002176677916180,
1.24227869265513002176677916180, 2.21815726173168531504182273995, 2.50203751714958064122409856717, 3.48355439196781874323725997784, 4.72173984764811909200463972071, 5.33295213934120228720826206170, 6.61487682809030095055817242167, 7.28034036221079194006370691208, 8.036484347716587976110533538713, 8.310709542379368773827746750368
