| L(s) = 1 | − 2-s − 3-s + 4-s − 0.662·5-s + 6-s − 0.537·7-s − 8-s + 9-s + 0.662·10-s + 1.91·11-s − 12-s − 2.29·13-s + 0.537·14-s + 0.662·15-s + 16-s − 1.24·17-s − 18-s + 2.15·19-s − 0.662·20-s + 0.537·21-s − 1.91·22-s + 23-s + 24-s − 4.56·25-s + 2.29·26-s − 27-s − 0.537·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.296·5-s + 0.408·6-s − 0.203·7-s − 0.353·8-s + 0.333·9-s + 0.209·10-s + 0.576·11-s − 0.288·12-s − 0.635·13-s + 0.143·14-s + 0.171·15-s + 0.250·16-s − 0.303·17-s − 0.235·18-s + 0.493·19-s − 0.148·20-s + 0.117·21-s − 0.407·22-s + 0.208·23-s + 0.204·24-s − 0.912·25-s + 0.449·26-s − 0.192·27-s − 0.101·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 + 0.662T + 5T^{2} \) |
| 7 | \( 1 + 0.537T + 7T^{2} \) |
| 11 | \( 1 - 1.91T + 11T^{2} \) |
| 13 | \( 1 + 2.29T + 13T^{2} \) |
| 17 | \( 1 + 1.24T + 17T^{2} \) |
| 19 | \( 1 - 2.15T + 19T^{2} \) |
| 31 | \( 1 - 4.51T + 31T^{2} \) |
| 37 | \( 1 + 4.68T + 37T^{2} \) |
| 41 | \( 1 - 3.06T + 41T^{2} \) |
| 43 | \( 1 - 3.78T + 43T^{2} \) |
| 47 | \( 1 + 3.25T + 47T^{2} \) |
| 53 | \( 1 + 1.37T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 - 0.986T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 - 4.39T + 71T^{2} \) |
| 73 | \( 1 - 6.04T + 73T^{2} \) |
| 79 | \( 1 - 0.602T + 79T^{2} \) |
| 83 | \( 1 + 5.40T + 83T^{2} \) |
| 89 | \( 1 + 1.27T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
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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
−8.049409319528013506824475400728, −7.35600982146380516398334985665, −6.71282273228537161894178353572, −6.03977870881608757263508343353, −5.15764514691083298485065694326, −4.29636515836982978500304164785, −3.36680924151244308604155666244, −2.29687088851393736990536033869, −1.17405488524144781838885095248, 0,
1.17405488524144781838885095248, 2.29687088851393736990536033869, 3.36680924151244308604155666244, 4.29636515836982978500304164785, 5.15764514691083298485065694326, 6.03977870881608757263508343353, 6.71282273228537161894178353572, 7.35600982146380516398334985665, 8.049409319528013506824475400728
