| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 7-s − 8-s + 9-s − 12-s − 2·13-s − 14-s + 16-s + 2·17-s − 18-s + 19-s − 21-s − 4·23-s + 24-s + 2·26-s − 27-s + 28-s − 2·29-s − 32-s − 2·34-s + 36-s + 2·37-s − 38-s + 2·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.288·12-s − 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.229·19-s − 0.218·21-s − 0.834·23-s + 0.204·24-s + 0.392·26-s − 0.192·27-s + 0.188·28-s − 0.371·29-s − 0.176·32-s − 0.342·34-s + 1/6·36-s + 0.328·37-s − 0.162·38-s + 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.14709561791500, −15.45679008127047, −14.94811480126010, −14.45016262781887, −13.75431159210079, −13.16388510185633, −12.42755281409553, −11.95253272754291, −11.54837243233859, −10.97045605834139, −10.19385865767907, −10.02336151353439, −9.298177502773601, −8.623512375094879, −7.994504234235591, −7.511244701993201, −6.873214722886031, −6.290830239303340, −5.492259388063058, −5.118877991615428, −4.200132356436926, −3.513472915360071, −2.535674006301408, −1.823645380840249, −0.9718945048863088, 0,
0.9718945048863088, 1.823645380840249, 2.535674006301408, 3.513472915360071, 4.200132356436926, 5.118877991615428, 5.492259388063058, 6.290830239303340, 6.873214722886031, 7.511244701993201, 7.994504234235591, 8.623512375094879, 9.298177502773601, 10.02336151353439, 10.19385865767907, 10.97045605834139, 11.54837243233859, 11.95253272754291, 12.42755281409553, 13.16388510185633, 13.75431159210079, 14.45016262781887, 14.94811480126010, 15.45679008127047, 16.14709561791500
