| L(s) = 1 | − 2-s + 3-s + 4-s + 2·5-s − 6-s − 8-s + 9-s − 2·10-s + 4·11-s + 12-s − 2·13-s + 2·15-s + 16-s + 2·17-s − 18-s − 4·19-s + 2·20-s − 4·22-s + 4·23-s − 24-s − 25-s + 2·26-s + 27-s − 2·29-s − 2·30-s − 32-s + 4·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 1.20·11-s + 0.288·12-s − 0.554·13-s + 0.516·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.917·19-s + 0.447·20-s − 0.852·22-s + 0.834·23-s − 0.204·24-s − 1/5·25-s + 0.392·26-s + 0.192·27-s − 0.371·29-s − 0.365·30-s − 0.176·32-s + 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 402 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 402 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.447247950\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.447247950\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
| 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
−11.06223046667890120446083698663, −10.07899863590022901186798096007, −9.378370539443186794864529493265, −8.782877656823769658320750680612, −7.63396440183345063471670724764, −6.69359428565717406351438852151, −5.74039868256347747327955323455, −4.19907474425947533976280457040, −2.72165748573829794861495187633, −1.50955704520556597491566025197,
1.50955704520556597491566025197, 2.72165748573829794861495187633, 4.19907474425947533976280457040, 5.74039868256347747327955323455, 6.69359428565717406351438852151, 7.63396440183345063471670724764, 8.782877656823769658320750680612, 9.378370539443186794864529493265, 10.07899863590022901186798096007, 11.06223046667890120446083698663
