| L(s) = 1 | + 2.89·3-s + 0.580·7-s + 5.40·9-s − 0.0809·11-s + 3.02·13-s − 0.280·17-s + 4.72·19-s + 1.68·21-s − 23-s + 6.98·27-s + 1.38·29-s − 5.70·31-s − 0.234·33-s − 2.61·37-s + 8.75·39-s + 5.31·41-s + 7.30·43-s + 12.2·47-s − 6.66·49-s − 0.813·51-s − 6.64·53-s + 13.7·57-s + 3.70·59-s + 11.5·61-s + 3.14·63-s − 5.99·67-s − 2.89·69-s + ⋯ |
| L(s) = 1 | + 1.67·3-s + 0.219·7-s + 1.80·9-s − 0.0243·11-s + 0.837·13-s − 0.0680·17-s + 1.08·19-s + 0.367·21-s − 0.208·23-s + 1.34·27-s + 0.256·29-s − 1.02·31-s − 0.0408·33-s − 0.429·37-s + 1.40·39-s + 0.830·41-s + 1.11·43-s + 1.78·47-s − 0.951·49-s − 0.113·51-s − 0.912·53-s + 1.81·57-s + 0.481·59-s + 1.47·61-s + 0.395·63-s − 0.732·67-s − 0.349·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.174106968\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.174106968\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 2.89T + 3T^{2} \) |
| 7 | \( 1 - 0.580T + 7T^{2} \) |
| 11 | \( 1 + 0.0809T + 11T^{2} \) |
| 13 | \( 1 - 3.02T + 13T^{2} \) |
| 17 | \( 1 + 0.280T + 17T^{2} \) |
| 19 | \( 1 - 4.72T + 19T^{2} \) |
| 29 | \( 1 - 1.38T + 29T^{2} \) |
| 31 | \( 1 + 5.70T + 31T^{2} \) |
| 37 | \( 1 + 2.61T + 37T^{2} \) |
| 41 | \( 1 - 5.31T + 41T^{2} \) |
| 43 | \( 1 - 7.30T + 43T^{2} \) |
| 47 | \( 1 - 12.2T + 47T^{2} \) |
| 53 | \( 1 + 6.64T + 53T^{2} \) |
| 59 | \( 1 - 3.70T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 + 5.99T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 - 1.14T + 73T^{2} \) |
| 79 | \( 1 + 1.68T + 79T^{2} \) |
| 83 | \( 1 - 9.86T + 83T^{2} \) |
| 89 | \( 1 - 6.69T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
| 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
−8.398868247382385687328029540421, −7.61562453329286588661456374574, −7.28398877686637596345388947615, −6.19781310569297431856128385434, −5.33576410243019265418254039488, −4.27429973120190300424814874400, −3.65221957120167180489698578458, −2.92926237233345433231470239688, −2.08578958360546987179128165817, −1.14095780698926748061956410108,
1.14095780698926748061956410108, 2.08578958360546987179128165817, 2.92926237233345433231470239688, 3.65221957120167180489698578458, 4.27429973120190300424814874400, 5.33576410243019265418254039488, 6.19781310569297431856128385434, 7.28398877686637596345388947615, 7.61562453329286588661456374574, 8.398868247382385687328029540421
