| L(s) = 1 | + 2-s − 1.80·3-s + 4-s + 0.856·5-s − 1.80·6-s + 0.353·7-s + 8-s + 0.274·9-s + 0.856·10-s − 2.40·11-s − 1.80·12-s + 2.50·13-s + 0.353·14-s − 1.55·15-s + 16-s − 2.24·17-s + 0.274·18-s + 3.35·19-s + 0.856·20-s − 0.639·21-s − 2.40·22-s − 6.48·23-s − 1.80·24-s − 4.26·25-s + 2.50·26-s + 4.93·27-s + 0.353·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.04·3-s + 0.5·4-s + 0.383·5-s − 0.738·6-s + 0.133·7-s + 0.353·8-s + 0.0915·9-s + 0.270·10-s − 0.724·11-s − 0.522·12-s + 0.694·13-s + 0.0944·14-s − 0.400·15-s + 0.250·16-s − 0.544·17-s + 0.0647·18-s + 0.769·19-s + 0.191·20-s − 0.139·21-s − 0.512·22-s − 1.35·23-s − 0.369·24-s − 0.853·25-s + 0.491·26-s + 0.949·27-s + 0.0668·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 2017 | \( 1 - T \) |
| good | 3 | \( 1 + 1.80T + 3T^{2} \) |
| 5 | \( 1 - 0.856T + 5T^{2} \) |
| 7 | \( 1 - 0.353T + 7T^{2} \) |
| 11 | \( 1 + 2.40T + 11T^{2} \) |
| 13 | \( 1 - 2.50T + 13T^{2} \) |
| 17 | \( 1 + 2.24T + 17T^{2} \) |
| 19 | \( 1 - 3.35T + 19T^{2} \) |
| 23 | \( 1 + 6.48T + 23T^{2} \) |
| 29 | \( 1 + 1.23T + 29T^{2} \) |
| 31 | \( 1 - 2.75T + 31T^{2} \) |
| 37 | \( 1 - 0.534T + 37T^{2} \) |
| 41 | \( 1 + 5.25T + 41T^{2} \) |
| 43 | \( 1 + 12.6T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 + 5.22T + 53T^{2} \) |
| 59 | \( 1 - 7.53T + 59T^{2} \) |
| 61 | \( 1 - 8.73T + 61T^{2} \) |
| 67 | \( 1 - 2.48T + 67T^{2} \) |
| 71 | \( 1 + 8.37T + 71T^{2} \) |
| 73 | \( 1 + 6.15T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 + 2.65T + 83T^{2} \) |
| 89 | \( 1 + 0.509T + 89T^{2} \) |
| 97 | \( 1 + 15.7T + 97T^{2} \) |
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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
−8.027715263414211404501553186450, −7.07135598919725979851019308887, −6.33448351448431505367391705537, −5.74170711992358117525715118086, −5.26946196819193297515666227058, −4.44876763576176578636552681718, −3.53805067795220535709843154568, −2.51203884205864031675006193642, −1.48606532015264564404090822924, 0,
1.48606532015264564404090822924, 2.51203884205864031675006193642, 3.53805067795220535709843154568, 4.44876763576176578636552681718, 5.26946196819193297515666227058, 5.74170711992358117525715118086, 6.33448351448431505367391705537, 7.07135598919725979851019308887, 8.027715263414211404501553186450
