| L(s) = 1 | − 1.31·2-s + 2.95·3-s − 0.279·4-s − 5-s − 3.87·6-s + 1.12·7-s + 2.98·8-s + 5.71·9-s + 1.31·10-s + 5.24·11-s − 0.825·12-s − 4.00·13-s − 1.46·14-s − 2.95·15-s − 3.36·16-s − 0.102·17-s − 7.50·18-s + 3.70·19-s + 0.279·20-s + 3.30·21-s − 6.88·22-s − 6.85·23-s + 8.82·24-s + 25-s + 5.25·26-s + 8.02·27-s − 0.313·28-s + ⋯ |
| L(s) = 1 | − 0.927·2-s + 1.70·3-s − 0.139·4-s − 0.447·5-s − 1.58·6-s + 0.423·7-s + 1.05·8-s + 1.90·9-s + 0.414·10-s + 1.58·11-s − 0.238·12-s − 1.11·13-s − 0.392·14-s − 0.762·15-s − 0.840·16-s − 0.0249·17-s − 1.76·18-s + 0.850·19-s + 0.0625·20-s + 0.721·21-s − 1.46·22-s − 1.43·23-s + 1.80·24-s + 0.200·25-s + 1.03·26-s + 1.54·27-s − 0.0591·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.944719447\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.944719447\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 + 1.31T + 2T^{2} \) |
| 3 | \( 1 - 2.95T + 3T^{2} \) |
| 7 | \( 1 - 1.12T + 7T^{2} \) |
| 11 | \( 1 - 5.24T + 11T^{2} \) |
| 13 | \( 1 + 4.00T + 13T^{2} \) |
| 17 | \( 1 + 0.102T + 17T^{2} \) |
| 19 | \( 1 - 3.70T + 19T^{2} \) |
| 23 | \( 1 + 6.85T + 23T^{2} \) |
| 29 | \( 1 - 2.01T + 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 - 2.90T + 37T^{2} \) |
| 41 | \( 1 - 11.8T + 41T^{2} \) |
| 43 | \( 1 - 3.80T + 43T^{2} \) |
| 47 | \( 1 - 0.424T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 + 9.06T + 59T^{2} \) |
| 61 | \( 1 + 10.7T + 61T^{2} \) |
| 67 | \( 1 - 14.8T + 67T^{2} \) |
| 71 | \( 1 - 9.72T + 71T^{2} \) |
| 73 | \( 1 + 7.13T + 73T^{2} \) |
| 79 | \( 1 + 7.65T + 79T^{2} \) |
| 83 | \( 1 + 8.28T + 83T^{2} \) |
| 89 | \( 1 - 9.60T + 89T^{2} \) |
| 97 | \( 1 - 5.34T + 97T^{2} \) |
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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
−9.214744900841936055628839504140, −8.364601526722852510957594714744, −7.83167707070056077047992036973, −7.41102531687272145671994407187, −6.34121685415468379626152487750, −4.59303365924669249508199099144, −4.23991822584603471970989772108, −3.17288356415062750171476895170, −2.08456251434778131968227503051, −1.07434603086066415057974011315,
1.07434603086066415057974011315, 2.08456251434778131968227503051, 3.17288356415062750171476895170, 4.23991822584603471970989772108, 4.59303365924669249508199099144, 6.34121685415468379626152487750, 7.41102531687272145671994407187, 7.83167707070056077047992036973, 8.364601526722852510957594714744, 9.214744900841936055628839504140
