L(s) = 1 | − 3-s + 5-s + 7-s + 9-s + 3.12·11-s − 5.12·13-s − 15-s + 2·17-s − 7.12·19-s − 21-s + 25-s − 27-s + 2·29-s − 3.12·31-s − 3.12·33-s + 35-s + 2·37-s + 5.12·39-s + 2·41-s + 6.24·43-s + 45-s + 49-s − 2·51-s + 11.3·53-s + 3.12·55-s + 7.12·57-s + 4·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 0.333·9-s + 0.941·11-s − 1.42·13-s − 0.258·15-s + 0.485·17-s − 1.63·19-s − 0.218·21-s + 0.200·25-s − 0.192·27-s + 0.371·29-s − 0.560·31-s − 0.543·33-s + 0.169·35-s + 0.328·37-s + 0.820·39-s + 0.312·41-s + 0.952·43-s + 0.149·45-s + 0.142·49-s − 0.280·51-s + 1.56·53-s + 0.421·55-s + 0.943·57-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.688502222\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.688502222\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
good | 11 | \( 1 - 3.12T + 11T^{2} \) |
| 13 | \( 1 + 5.12T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 7.12T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 3.12T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 6.24T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 8.24T + 61T^{2} \) |
| 67 | \( 1 + 14.2T + 67T^{2} \) |
| 71 | \( 1 - 3.12T + 71T^{2} \) |
| 73 | \( 1 - 6.87T + 73T^{2} \) |
| 79 | \( 1 - 14.2T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 16.2T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.82389758625058590301950783520, −7.25164144335442151100501031169, −6.46946987729039197030577977881, −5.96153218349929408049912422304, −5.09880520705935399400457081385, −4.51102712715630081017310728516, −3.77147443368869694533950532494, −2.52273585610845977470953891964, −1.84208102599593655266593948468, −0.68727139888052538684776912316,
0.68727139888052538684776912316, 1.84208102599593655266593948468, 2.52273585610845977470953891964, 3.77147443368869694533950532494, 4.51102712715630081017310728516, 5.09880520705935399400457081385, 5.96153218349929408049912422304, 6.46946987729039197030577977881, 7.25164144335442151100501031169, 7.82389758625058590301950783520