L(s) = 1 | − 3-s − 5-s + 7-s + 9-s − 13-s + 15-s + 6·17-s + 4·19-s − 21-s + 25-s − 27-s − 6·29-s − 4·31-s − 35-s + 10·37-s + 39-s + 6·41-s − 8·43-s − 45-s + 49-s − 6·51-s + 6·53-s − 4·57-s + 12·59-s − 14·61-s + 63-s + 65-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.277·13-s + 0.258·15-s + 1.45·17-s + 0.917·19-s − 0.218·21-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 0.718·31-s − 0.169·35-s + 1.64·37-s + 0.160·39-s + 0.937·41-s − 1.21·43-s − 0.149·45-s + 1/7·49-s − 0.840·51-s + 0.824·53-s − 0.529·57-s + 1.56·59-s − 1.79·61-s + 0.125·63-s + 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.035143623\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.035143623\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−13.88381859153427, −13.32509536617211, −12.89695512573916, −12.14931135108279, −12.07079831635317, −11.37115660742446, −11.10491388569920, −10.46629585290129, −9.938487081977243, −9.425771046265241, −9.050618960712025, −8.051950656097775, −7.863646917439189, −7.369463842575492, −6.842576709276161, −6.090592901910933, −5.549900379663710, −5.209990760336431, −4.578098959498429, −3.870699926453742, −3.446837005464409, −2.699849229531420, −1.894148237009959, −1.126566025715894, −0.5448283700703436,
0.5448283700703436, 1.126566025715894, 1.894148237009959, 2.699849229531420, 3.446837005464409, 3.870699926453742, 4.578098959498429, 5.209990760336431, 5.549900379663710, 6.090592901910933, 6.842576709276161, 7.369463842575492, 7.863646917439189, 8.051950656097775, 9.050618960712025, 9.425771046265241, 9.938487081977243, 10.46629585290129, 11.10491388569920, 11.37115660742446, 12.07079831635317, 12.14931135108279, 12.89695512573916, 13.32509536617211, 13.88381859153427