| L(s) = 1 | − 2-s − 3-s + 4-s − 0.427·5-s + 6-s − 1.56·7-s − 8-s + 9-s + 0.427·10-s + 4.81·11-s − 12-s − 0.388·13-s + 1.56·14-s + 0.427·15-s + 16-s + 4.71·17-s − 18-s − 19-s − 0.427·20-s + 1.56·21-s − 4.81·22-s − 4.41·23-s + 24-s − 4.81·25-s + 0.388·26-s − 27-s − 1.56·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.191·5-s + 0.408·6-s − 0.591·7-s − 0.353·8-s + 0.333·9-s + 0.135·10-s + 1.45·11-s − 0.288·12-s − 0.107·13-s + 0.418·14-s + 0.110·15-s + 0.250·16-s + 1.14·17-s − 0.235·18-s − 0.229·19-s − 0.0956·20-s + 0.341·21-s − 1.02·22-s − 0.920·23-s + 0.204·24-s − 0.963·25-s + 0.0761·26-s − 0.192·27-s − 0.295·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| good | 5 | \( 1 + 0.427T + 5T^{2} \) |
| 7 | \( 1 + 1.56T + 7T^{2} \) |
| 11 | \( 1 - 4.81T + 11T^{2} \) |
| 13 | \( 1 + 0.388T + 13T^{2} \) |
| 17 | \( 1 - 4.71T + 17T^{2} \) |
| 23 | \( 1 + 4.41T + 23T^{2} \) |
| 29 | \( 1 - 4.21T + 29T^{2} \) |
| 31 | \( 1 + 5.84T + 31T^{2} \) |
| 37 | \( 1 - 4.83T + 37T^{2} \) |
| 41 | \( 1 + 5.41T + 41T^{2} \) |
| 43 | \( 1 + 1.29T + 43T^{2} \) |
| 47 | \( 1 + 12.1T + 47T^{2} \) |
| 59 | \( 1 - 0.782T + 59T^{2} \) |
| 61 | \( 1 - 7.68T + 61T^{2} \) |
| 67 | \( 1 + 1.73T + 67T^{2} \) |
| 71 | \( 1 - 5.66T + 71T^{2} \) |
| 73 | \( 1 - 0.430T + 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 + 6.44T + 83T^{2} \) |
| 89 | \( 1 - 6.44T + 89T^{2} \) |
| 97 | \( 1 + 2.09T + 97T^{2} \) |
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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
−7.79659494453881706197215722025, −6.93110355107125612497542901572, −6.39181581699195549284537331208, −5.86195382960851844248676763250, −4.90032312837520797830118809856, −3.85404495373983601198730969130, −3.36660625733110962104302730918, −2.02698585620351945354424181219, −1.15297517597175311343071819847, 0,
1.15297517597175311343071819847, 2.02698585620351945354424181219, 3.36660625733110962104302730918, 3.85404495373983601198730969130, 4.90032312837520797830118809856, 5.86195382960851844248676763250, 6.39181581699195549284537331208, 6.93110355107125612497542901572, 7.79659494453881706197215722025
