| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 8-s + 9-s − 10-s − 12-s + 2·13-s + 15-s + 16-s + 18-s + 8·19-s − 20-s + 23-s − 24-s + 25-s + 2·26-s − 27-s + 30-s − 6·31-s + 32-s + 36-s + 10·37-s + 8·38-s − 2·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s + 0.554·13-s + 0.258·15-s + 1/4·16-s + 0.235·18-s + 1.83·19-s − 0.223·20-s + 0.208·23-s − 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s + 0.182·30-s − 1.07·31-s + 0.176·32-s + 1/6·36-s + 1.64·37-s + 1.29·38-s − 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 199410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.25126957995764, −12.86136220513374, −12.21270194769161, −12.02635061837684, −11.42720078172634, −11.03816195775398, −10.83130797043065, −10.00615680463905, −9.628648152503519, −9.146787693238729, −8.466247984229424, −7.861710406154538, −7.491170903626505, −7.051983346139670, −6.404587834973697, −6.054220547009346, −5.334396151633246, −5.146759809438717, −4.497104874881576, −3.901721639393588, −3.426082712982374, −2.973380463433448, −2.205662408737062, −1.380870888560577, −0.9521778845010082, 0,
0.9521778845010082, 1.380870888560577, 2.205662408737062, 2.973380463433448, 3.426082712982374, 3.901721639393588, 4.497104874881576, 5.146759809438717, 5.334396151633246, 6.054220547009346, 6.404587834973697, 7.051983346139670, 7.491170903626505, 7.861710406154538, 8.466247984229424, 9.146787693238729, 9.628648152503519, 10.00615680463905, 10.83130797043065, 11.03816195775398, 11.42720078172634, 12.02635061837684, 12.21270194769161, 12.86136220513374, 13.25126957995764
