| L(s) = 1 | + 2-s − 2.92·3-s + 4-s + 5-s − 2.92·6-s + 4.55·7-s + 8-s + 5.55·9-s + 10-s − 0.925·11-s − 2.92·12-s + 2.55·13-s + 4.55·14-s − 2.92·15-s + 16-s + 6.92·17-s + 5.55·18-s − 3.29·19-s + 20-s − 13.3·21-s − 0.925·22-s − 2.92·24-s + 25-s + 2.55·26-s − 7.48·27-s + 4.55·28-s − 1.26·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.68·3-s + 0.5·4-s + 0.447·5-s − 1.19·6-s + 1.72·7-s + 0.353·8-s + 1.85·9-s + 0.316·10-s − 0.279·11-s − 0.844·12-s + 0.709·13-s + 1.21·14-s − 0.755·15-s + 0.250·16-s + 1.67·17-s + 1.31·18-s − 0.755·19-s + 0.223·20-s − 2.90·21-s − 0.197·22-s − 0.597·24-s + 0.200·25-s + 0.501·26-s − 1.44·27-s + 0.861·28-s − 0.234·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.804954147\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.804954147\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 2.92T + 3T^{2} \) |
| 7 | \( 1 - 4.55T + 7T^{2} \) |
| 11 | \( 1 + 0.925T + 11T^{2} \) |
| 13 | \( 1 - 2.55T + 13T^{2} \) |
| 17 | \( 1 - 6.92T + 17T^{2} \) |
| 19 | \( 1 + 3.29T + 19T^{2} \) |
| 29 | \( 1 + 1.26T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 - 9.70T + 37T^{2} \) |
| 41 | \( 1 + 0.925T + 41T^{2} \) |
| 43 | \( 1 + 1.26T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 + 3.11T + 53T^{2} \) |
| 59 | \( 1 + 7.26T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 + 7.85T + 67T^{2} \) |
| 71 | \( 1 + 6.92T + 71T^{2} \) |
| 73 | \( 1 + 9.26T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 - 0.585T + 83T^{2} \) |
| 89 | \( 1 - 9.11T + 89T^{2} \) |
| 97 | \( 1 + 3.22T + 97T^{2} \) |
| 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
−7.85728435051426417229875790096, −7.45485677224112915112075176550, −6.28644802762068845977856989467, −5.94390351060001613395692275553, −5.34857419105031557573951247275, −4.62660706438408970557836459821, −4.24628067195662812871512533452, −2.83511571947649564628885518628, −1.58324526046119323904355867773, −1.01241983022320852744024051929,
1.01241983022320852744024051929, 1.58324526046119323904355867773, 2.83511571947649564628885518628, 4.24628067195662812871512533452, 4.62660706438408970557836459821, 5.34857419105031557573951247275, 5.94390351060001613395692275553, 6.28644802762068845977856989467, 7.45485677224112915112075176550, 7.85728435051426417229875790096
