| L(s) = 1 | + 2-s + 3-s + 4-s + 3·5-s + 6-s + 8-s + 9-s + 3·10-s + 4·11-s + 12-s + 3·13-s + 3·15-s + 16-s + 2·17-s + 18-s + 8·19-s + 3·20-s + 4·22-s + 4·23-s + 24-s + 4·25-s + 3·26-s + 27-s − 2·29-s + 3·30-s + 32-s + 4·33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.948·10-s + 1.20·11-s + 0.288·12-s + 0.832·13-s + 0.774·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 1.83·19-s + 0.670·20-s + 0.852·22-s + 0.834·23-s + 0.204·24-s + 4/5·25-s + 0.588·26-s + 0.192·27-s − 0.371·29-s + 0.547·30-s + 0.176·32-s + 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.60909108\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.60909108\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 131 | \( 1 \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 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 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
| 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
−13.71968071940693, −13.35626281458700, −12.94194927259450, −12.45035691074645, −11.75387778167364, −11.29880242907119, −11.00375244387188, −10.00505141480301, −9.789646001554496, −9.424580257439628, −8.862645308147276, −8.262323560870617, −7.683167356449466, −6.957467028277169, −6.654814508500835, −6.061786016713910, −5.495322668931704, −5.182696115044550, −4.397828489021019, −3.703861354884899, −3.310181306214111, −2.722970468223642, −2.009701610345926, −1.273994795054549, −1.078935990284552,
1.078935990284552, 1.273994795054549, 2.009701610345926, 2.722970468223642, 3.310181306214111, 3.703861354884899, 4.397828489021019, 5.182696115044550, 5.495322668931704, 6.061786016713910, 6.654814508500835, 6.957467028277169, 7.683167356449466, 8.262323560870617, 8.862645308147276, 9.424580257439628, 9.789646001554496, 10.00505141480301, 11.00375244387188, 11.29880242907119, 11.75387778167364, 12.45035691074645, 12.94194927259450, 13.35626281458700, 13.71968071940693
