| L(s) = 1 | − 2-s − 2.23·3-s + 4-s − 2.75·5-s + 2.23·6-s − 2.11·7-s − 8-s + 1.98·9-s + 2.75·10-s − 1.85·11-s − 2.23·12-s − 5.20·13-s + 2.11·14-s + 6.14·15-s + 16-s + 3.15·17-s − 1.98·18-s − 7.18·19-s − 2.75·20-s + 4.71·21-s + 1.85·22-s + 5.98·23-s + 2.23·24-s + 2.57·25-s + 5.20·26-s + 2.27·27-s − 2.11·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.28·3-s + 0.5·4-s − 1.23·5-s + 0.911·6-s − 0.797·7-s − 0.353·8-s + 0.660·9-s + 0.870·10-s − 0.558·11-s − 0.644·12-s − 1.44·13-s + 0.563·14-s + 1.58·15-s + 0.250·16-s + 0.765·17-s − 0.467·18-s − 1.64·19-s − 0.615·20-s + 1.02·21-s + 0.395·22-s + 1.24·23-s + 0.455·24-s + 0.515·25-s + 1.02·26-s + 0.437·27-s − 0.398·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2003 | \( 1 + T \) |
| good | 3 | \( 1 + 2.23T + 3T^{2} \) |
| 5 | \( 1 + 2.75T + 5T^{2} \) |
| 7 | \( 1 + 2.11T + 7T^{2} \) |
| 11 | \( 1 + 1.85T + 11T^{2} \) |
| 13 | \( 1 + 5.20T + 13T^{2} \) |
| 17 | \( 1 - 3.15T + 17T^{2} \) |
| 19 | \( 1 + 7.18T + 19T^{2} \) |
| 23 | \( 1 - 5.98T + 23T^{2} \) |
| 29 | \( 1 - 0.982T + 29T^{2} \) |
| 31 | \( 1 - 8.19T + 31T^{2} \) |
| 37 | \( 1 - 4.83T + 37T^{2} \) |
| 41 | \( 1 - 1.04T + 41T^{2} \) |
| 43 | \( 1 + 0.753T + 43T^{2} \) |
| 47 | \( 1 + 3.72T + 47T^{2} \) |
| 53 | \( 1 + 5.58T + 53T^{2} \) |
| 59 | \( 1 + 8.16T + 59T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 + 0.0465T + 71T^{2} \) |
| 73 | \( 1 + 3.43T + 73T^{2} \) |
| 79 | \( 1 - 0.984T + 79T^{2} \) |
| 83 | \( 1 + 1.05T + 83T^{2} \) |
| 89 | \( 1 + 4.09T + 89T^{2} \) |
| 97 | \( 1 - 11.5T + 97T^{2} \) |
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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
−8.016470787748025994429318637497, −7.33362085797342885154895875996, −6.65991349387488779716414593041, −6.09246885083562573621587063929, −5.02389404818090466657905700902, −4.51517092359100714467764855955, −3.31128680363656395966674892820, −2.47460413215811177740536119018, −0.74435366245040824810840571940, 0,
0.74435366245040824810840571940, 2.47460413215811177740536119018, 3.31128680363656395966674892820, 4.51517092359100714467764855955, 5.02389404818090466657905700902, 6.09246885083562573621587063929, 6.65991349387488779716414593041, 7.33362085797342885154895875996, 8.016470787748025994429318637497
