| L(s) = 1 | − 1.74·2-s − 3.34·3-s + 1.02·4-s − 2.66·5-s + 5.81·6-s + 0.0455·7-s + 1.68·8-s + 8.16·9-s + 4.63·10-s − 5.77·11-s − 3.44·12-s + 5.44·13-s − 0.0793·14-s + 8.90·15-s − 4.99·16-s − 2.61·17-s − 14.2·18-s + 19-s − 2.74·20-s − 0.152·21-s + 10.0·22-s − 5.93·23-s − 5.64·24-s + 2.10·25-s − 9.47·26-s − 17.2·27-s + 0.0469·28-s + ⋯ |
| L(s) = 1 | − 1.23·2-s − 1.92·3-s + 0.514·4-s − 1.19·5-s + 2.37·6-s + 0.0172·7-s + 0.597·8-s + 2.72·9-s + 1.46·10-s − 1.74·11-s − 0.993·12-s + 1.50·13-s − 0.0211·14-s + 2.29·15-s − 1.24·16-s − 0.634·17-s − 3.34·18-s + 0.229·19-s − 0.613·20-s − 0.0332·21-s + 2.14·22-s − 1.23·23-s − 1.15·24-s + 0.421·25-s − 1.85·26-s − 3.32·27-s + 0.00886·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
| good | 2 | \( 1 + 1.74T + 2T^{2} \) |
| 3 | \( 1 + 3.34T + 3T^{2} \) |
| 5 | \( 1 + 2.66T + 5T^{2} \) |
| 7 | \( 1 - 0.0455T + 7T^{2} \) |
| 11 | \( 1 + 5.77T + 11T^{2} \) |
| 13 | \( 1 - 5.44T + 13T^{2} \) |
| 17 | \( 1 + 2.61T + 17T^{2} \) |
| 23 | \( 1 + 5.93T + 23T^{2} \) |
| 29 | \( 1 + 1.01T + 29T^{2} \) |
| 31 | \( 1 - 5.87T + 31T^{2} \) |
| 37 | \( 1 + 5.35T + 37T^{2} \) |
| 41 | \( 1 + 6.75T + 41T^{2} \) |
| 43 | \( 1 - 8.74T + 43T^{2} \) |
| 47 | \( 1 + 9.44T + 47T^{2} \) |
| 53 | \( 1 - 0.930T + 53T^{2} \) |
| 59 | \( 1 - 6.84T + 59T^{2} \) |
| 61 | \( 1 - 3.42T + 61T^{2} \) |
| 67 | \( 1 - 12.3T + 67T^{2} \) |
| 71 | \( 1 - 1.90T + 71T^{2} \) |
| 73 | \( 1 - 0.651T + 73T^{2} \) |
| 79 | \( 1 + 8.80T + 79T^{2} \) |
| 83 | \( 1 + 2.19T + 83T^{2} \) |
| 89 | \( 1 + 5.58T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−8.099167909831522093514470969335, −7.45995394726275363964682812307, −6.73582044603807036871218106578, −5.97861174526427695152224733735, −5.11163026020261378755236517489, −4.46999130764678062620717893548, −3.65211848026746813308822585623, −1.89328848939447725188338661251, −0.72226416423688329413442254932, 0,
0.72226416423688329413442254932, 1.89328848939447725188338661251, 3.65211848026746813308822585623, 4.46999130764678062620717893548, 5.11163026020261378755236517489, 5.97861174526427695152224733735, 6.73582044603807036871218106578, 7.45995394726275363964682812307, 8.099167909831522093514470969335
