| L(s) = 1 | − 2-s + 4-s − 2·5-s − 2·7-s − 8-s − 3·9-s + 2·10-s − 2·11-s + 6·13-s + 2·14-s + 16-s − 17-s + 3·18-s − 2·20-s + 2·22-s + 6·23-s − 25-s − 6·26-s − 2·28-s − 32-s + 34-s + 4·35-s − 3·36-s − 8·37-s + 2·40-s − 4·43-s − 2·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.755·7-s − 0.353·8-s − 9-s + 0.632·10-s − 0.603·11-s + 1.66·13-s + 0.534·14-s + 1/4·16-s − 0.242·17-s + 0.707·18-s − 0.447·20-s + 0.426·22-s + 1.25·23-s − 1/5·25-s − 1.17·26-s − 0.377·28-s − 0.176·32-s + 0.171·34-s + 0.676·35-s − 1/2·36-s − 1.31·37-s + 0.316·40-s − 0.609·43-s − 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4881954652\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4881954652\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−16.32622417496580, −15.88121237470180, −15.38895130527436, −14.89699066564274, −14.09151453437607, −13.34762690210674, −13.02794117391923, −12.20553562118613, −11.56717979289760, −11.12032656745893, −10.73296928438662, −9.953394196838134, −9.242035058308841, −8.597944681429703, −8.298933813085909, −7.666234236032846, −6.753196792195048, −6.428637394331544, −5.579681598383717, −4.912603664135547, −3.668959528582603, −3.401005448444380, −2.615966698872677, −1.461531447781417, −0.3603451076063961,
0.3603451076063961, 1.461531447781417, 2.615966698872677, 3.401005448444380, 3.668959528582603, 4.912603664135547, 5.579681598383717, 6.428637394331544, 6.753196792195048, 7.666234236032846, 8.298933813085909, 8.597944681429703, 9.242035058308841, 9.953394196838134, 10.73296928438662, 11.12032656745893, 11.56717979289760, 12.20553562118613, 13.02794117391923, 13.34762690210674, 14.09151453437607, 14.89699066564274, 15.38895130527436, 15.88121237470180, 16.32622417496580
