| L(s) = 1 | − 1.57·2-s − 3-s + 0.477·4-s + 0.564·5-s + 1.57·6-s − 3.58·7-s + 2.39·8-s + 9-s − 0.888·10-s − 0.304·11-s − 0.477·12-s − 3.30·13-s + 5.63·14-s − 0.564·15-s − 4.72·16-s + 5.79·17-s − 1.57·18-s + 3.49·19-s + 0.269·20-s + 3.58·21-s + 0.478·22-s − 23-s − 2.39·24-s − 4.68·25-s + 5.20·26-s − 27-s − 1.71·28-s + ⋯ |
| L(s) = 1 | − 1.11·2-s − 0.577·3-s + 0.238·4-s + 0.252·5-s + 0.642·6-s − 1.35·7-s + 0.847·8-s + 0.333·9-s − 0.281·10-s − 0.0917·11-s − 0.137·12-s − 0.917·13-s + 1.50·14-s − 0.145·15-s − 1.18·16-s + 1.40·17-s − 0.371·18-s + 0.801·19-s + 0.0603·20-s + 0.781·21-s + 0.102·22-s − 0.208·23-s − 0.489·24-s − 0.936·25-s + 1.02·26-s − 0.192·27-s − 0.323·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 1.57T + 2T^{2} \) |
| 5 | \( 1 - 0.564T + 5T^{2} \) |
| 7 | \( 1 + 3.58T + 7T^{2} \) |
| 11 | \( 1 + 0.304T + 11T^{2} \) |
| 13 | \( 1 + 3.30T + 13T^{2} \) |
| 17 | \( 1 - 5.79T + 17T^{2} \) |
| 19 | \( 1 - 3.49T + 19T^{2} \) |
| 31 | \( 1 - 2.40T + 31T^{2} \) |
| 37 | \( 1 - 6.97T + 37T^{2} \) |
| 41 | \( 1 - 0.962T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 47 | \( 1 + 7.76T + 47T^{2} \) |
| 53 | \( 1 - 6.81T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 + 9.74T + 61T^{2} \) |
| 67 | \( 1 - 0.430T + 67T^{2} \) |
| 71 | \( 1 + 7.39T + 71T^{2} \) |
| 73 | \( 1 - 9.36T + 73T^{2} \) |
| 79 | \( 1 - 1.31T + 79T^{2} \) |
| 83 | \( 1 - 0.543T + 83T^{2} \) |
| 89 | \( 1 - 10.0T + 89T^{2} \) |
| 97 | \( 1 - 4.33T + 97T^{2} \) |
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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
−9.127204699459798105115387697713, −7.69715849995846117117613192233, −7.60583478343024736685017856141, −6.45161378698747614851978153987, −5.75847880034283833675991767678, −4.82229316420148767254696468945, −3.70336006454719280512772394098, −2.57698184202001642241448106618, −1.14395145359265771726957820746, 0,
1.14395145359265771726957820746, 2.57698184202001642241448106618, 3.70336006454719280512772394098, 4.82229316420148767254696468945, 5.75847880034283833675991767678, 6.45161378698747614851978153987, 7.60583478343024736685017856141, 7.69715849995846117117613192233, 9.127204699459798105115387697713
