| L(s) = 1 | + 2-s + 2.64·3-s + 4-s − 1.64·5-s + 2.64·6-s + 3.64·7-s + 8-s + 4.00·9-s − 1.64·10-s − 4.64·11-s + 2.64·12-s + 2·13-s + 3.64·14-s − 4.35·15-s + 16-s + 4.00·18-s − 1.64·20-s + 9.64·21-s − 4.64·22-s − 1.64·23-s + 2.64·24-s − 2.29·25-s + 2·26-s + 2.64·27-s + 3.64·28-s + 1.64·29-s − 4.35·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.52·3-s + 0.5·4-s − 0.736·5-s + 1.08·6-s + 1.37·7-s + 0.353·8-s + 1.33·9-s − 0.520·10-s − 1.40·11-s + 0.763·12-s + 0.554·13-s + 0.974·14-s − 1.12·15-s + 0.250·16-s + 0.942·18-s − 0.368·20-s + 2.10·21-s − 0.990·22-s − 0.343·23-s + 0.540·24-s − 0.458·25-s + 0.392·26-s + 0.509·27-s + 0.688·28-s + 0.305·29-s − 0.794·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.601938840\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.601938840\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 2.64T + 3T^{2} \) |
| 5 | \( 1 + 1.64T + 5T^{2} \) |
| 7 | \( 1 - 3.64T + 7T^{2} \) |
| 11 | \( 1 + 4.64T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 23 | \( 1 + 1.64T + 23T^{2} \) |
| 29 | \( 1 - 1.64T + 29T^{2} \) |
| 31 | \( 1 + 5.64T + 31T^{2} \) |
| 37 | \( 1 - 0.354T + 37T^{2} \) |
| 41 | \( 1 + 0.291T + 41T^{2} \) |
| 43 | \( 1 - 11.2T + 43T^{2} \) |
| 47 | \( 1 - 4.35T + 47T^{2} \) |
| 53 | \( 1 + 12.5T + 53T^{2} \) |
| 59 | \( 1 + 7.93T + 59T^{2} \) |
| 61 | \( 1 - 0.937T + 61T^{2} \) |
| 67 | \( 1 - 0.645T + 67T^{2} \) |
| 71 | \( 1 + 2.70T + 71T^{2} \) |
| 73 | \( 1 - 1.70T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 7.93T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 3.70T + 97T^{2} \) |
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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
−10.64725024500250855364865192482, −9.369242086241083395490792602705, −8.322436675843627461357544597369, −7.892085565161539656560118929761, −7.36206897853501365838996878174, −5.72977802019590676732568321688, −4.66266035062607839491231076587, −3.86153329794411514718633421731, −2.83206089925943182489604862133, −1.83335734366817832146262396792,
1.83335734366817832146262396792, 2.83206089925943182489604862133, 3.86153329794411514718633421731, 4.66266035062607839491231076587, 5.72977802019590676732568321688, 7.36206897853501365838996878174, 7.892085565161539656560118929761, 8.322436675843627461357544597369, 9.369242086241083395490792602705, 10.64725024500250855364865192482
