| L(s) = 1 | − 2-s + 0.941·3-s + 4-s − 1.56·5-s − 0.941·6-s + 3.96·7-s − 8-s − 2.11·9-s + 1.56·10-s + 0.941·12-s − 2.44·13-s − 3.96·14-s − 1.47·15-s + 16-s − 1.45·17-s + 2.11·18-s − 2.84·19-s − 1.56·20-s + 3.72·21-s − 23-s − 0.941·24-s − 2.53·25-s + 2.44·26-s − 4.81·27-s + 3.96·28-s + 5.71·29-s + 1.47·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.543·3-s + 0.5·4-s − 0.702·5-s − 0.384·6-s + 1.49·7-s − 0.353·8-s − 0.704·9-s + 0.496·10-s + 0.271·12-s − 0.678·13-s − 1.05·14-s − 0.381·15-s + 0.250·16-s − 0.352·17-s + 0.498·18-s − 0.653·19-s − 0.351·20-s + 0.813·21-s − 0.208·23-s − 0.192·24-s − 0.507·25-s + 0.480·26-s − 0.926·27-s + 0.748·28-s + 1.06·29-s + 0.269·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5566 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5566 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.376215723\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.376215723\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 0.941T + 3T^{2} \) |
| 5 | \( 1 + 1.56T + 5T^{2} \) |
| 7 | \( 1 - 3.96T + 7T^{2} \) |
| 13 | \( 1 + 2.44T + 13T^{2} \) |
| 17 | \( 1 + 1.45T + 17T^{2} \) |
| 19 | \( 1 + 2.84T + 19T^{2} \) |
| 29 | \( 1 - 5.71T + 29T^{2} \) |
| 31 | \( 1 - 1.90T + 31T^{2} \) |
| 37 | \( 1 - 3.57T + 37T^{2} \) |
| 41 | \( 1 + 1.83T + 41T^{2} \) |
| 43 | \( 1 + 5.06T + 43T^{2} \) |
| 47 | \( 1 - 6.81T + 47T^{2} \) |
| 53 | \( 1 - 14.3T + 53T^{2} \) |
| 59 | \( 1 - 0.992T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 - 7.10T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 + 0.691T + 73T^{2} \) |
| 79 | \( 1 - 9.17T + 79T^{2} \) |
| 83 | \( 1 - 9.92T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 + 19.3T + 97T^{2} \) |
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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
−8.115626389920427948172637396005, −7.81067850566231258005846177154, −7.01850174378116210592307646624, −6.11041759829965838463712112827, −5.16824883883953882508750767602, −4.46838300006352509590490205916, −3.62640309874311328493687217172, −2.51358349808823917946644265639, −1.98863561456491583546233998145, −0.66241330364314865676804287016,
0.66241330364314865676804287016, 1.98863561456491583546233998145, 2.51358349808823917946644265639, 3.62640309874311328493687217172, 4.46838300006352509590490205916, 5.16824883883953882508750767602, 6.11041759829965838463712112827, 7.01850174378116210592307646624, 7.81067850566231258005846177154, 8.115626389920427948172637396005
