| L(s) = 1 | − 2-s + 4-s − 8-s + 2·13-s + 16-s − 2·19-s − 2·26-s − 6·29-s − 8·31-s − 32-s + 4·37-s + 2·38-s + 6·41-s − 2·43-s + 6·47-s + 2·52-s + 6·53-s + 6·58-s + 12·59-s − 8·61-s + 8·62-s + 64-s − 2·67-s − 6·71-s + 2·73-s − 4·74-s − 2·76-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 0.554·13-s + 1/4·16-s − 0.458·19-s − 0.392·26-s − 1.11·29-s − 1.43·31-s − 0.176·32-s + 0.657·37-s + 0.324·38-s + 0.937·41-s − 0.304·43-s + 0.875·47-s + 0.277·52-s + 0.824·53-s + 0.787·58-s + 1.56·59-s − 1.02·61-s + 1.01·62-s + 1/8·64-s − 0.244·67-s − 0.712·71-s + 0.234·73-s − 0.464·74-s − 0.229·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.245858826\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.245858826\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
| 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
−15.57494584346243, −15.03378599425787, −14.60089998581392, −13.99371424718268, −13.16105869777569, −12.92379380595004, −12.21679380482197, −11.48330951888444, −11.16276583337846, −10.53026574881599, −10.07207400338930, −9.213561315605073, −9.020945816725355, −8.323911539293673, −7.646401132098773, −7.215136717104287, −6.532447507036813, −5.781320913188109, −5.447561559338884, −4.324548065367914, −3.831356851248642, −2.986082253837485, −2.187589924429821, −1.490842725816866, −0.5139052543364689,
0.5139052543364689, 1.490842725816866, 2.187589924429821, 2.986082253837485, 3.831356851248642, 4.324548065367914, 5.447561559338884, 5.781320913188109, 6.532447507036813, 7.215136717104287, 7.646401132098773, 8.323911539293673, 9.020945816725355, 9.213561315605073, 10.07207400338930, 10.53026574881599, 11.16276583337846, 11.48330951888444, 12.21679380482197, 12.92379380595004, 13.16105869777569, 13.99371424718268, 14.60089998581392, 15.03378599425787, 15.57494584346243
