| L(s) = 1 | − 2-s + 3.14·3-s + 4-s + 1.25·5-s − 3.14·6-s + 0.693·7-s − 8-s + 6.88·9-s − 1.25·10-s − 2.96·11-s + 3.14·12-s − 3.04·13-s − 0.693·14-s + 3.96·15-s + 16-s + 5.27·17-s − 6.88·18-s − 1.99·19-s + 1.25·20-s + 2.18·21-s + 2.96·22-s − 0.617·23-s − 3.14·24-s − 3.41·25-s + 3.04·26-s + 12.2·27-s + 0.693·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.81·3-s + 0.5·4-s + 0.563·5-s − 1.28·6-s + 0.262·7-s − 0.353·8-s + 2.29·9-s − 0.398·10-s − 0.892·11-s + 0.907·12-s − 0.844·13-s − 0.185·14-s + 1.02·15-s + 0.250·16-s + 1.27·17-s − 1.62·18-s − 0.456·19-s + 0.281·20-s + 0.475·21-s + 0.631·22-s − 0.128·23-s − 0.641·24-s − 0.682·25-s + 0.597·26-s + 2.35·27-s + 0.131·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.165921468\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.165921468\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2011 | \( 1 - T \) |
| good | 3 | \( 1 - 3.14T + 3T^{2} \) |
| 5 | \( 1 - 1.25T + 5T^{2} \) |
| 7 | \( 1 - 0.693T + 7T^{2} \) |
| 11 | \( 1 + 2.96T + 11T^{2} \) |
| 13 | \( 1 + 3.04T + 13T^{2} \) |
| 17 | \( 1 - 5.27T + 17T^{2} \) |
| 19 | \( 1 + 1.99T + 19T^{2} \) |
| 23 | \( 1 + 0.617T + 23T^{2} \) |
| 29 | \( 1 - 2.75T + 29T^{2} \) |
| 31 | \( 1 - 6.02T + 31T^{2} \) |
| 37 | \( 1 - 6.10T + 37T^{2} \) |
| 41 | \( 1 - 9.58T + 41T^{2} \) |
| 43 | \( 1 - 3.18T + 43T^{2} \) |
| 47 | \( 1 + 3.51T + 47T^{2} \) |
| 53 | \( 1 - 5.03T + 53T^{2} \) |
| 59 | \( 1 + 3.73T + 59T^{2} \) |
| 61 | \( 1 - 13.1T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 + 5.24T + 73T^{2} \) |
| 79 | \( 1 - 3.15T + 79T^{2} \) |
| 83 | \( 1 - 4.00T + 83T^{2} \) |
| 89 | \( 1 - 1.17T + 89T^{2} \) |
| 97 | \( 1 - 15.5T + 97T^{2} \) |
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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
−8.192150902356345798237889214580, −8.029118439695131344563334072106, −7.42103533161539622411120900403, −6.51188227797936022541228605522, −5.48942758049191367447940901631, −4.52356607275079697081928298326, −3.53534035522008804026867483261, −2.52746378536143578466743687307, −2.30917423890665700453061190031, −1.08658386956718830546075633549,
1.08658386956718830546075633549, 2.30917423890665700453061190031, 2.52746378536143578466743687307, 3.53534035522008804026867483261, 4.52356607275079697081928298326, 5.48942758049191367447940901631, 6.51188227797936022541228605522, 7.42103533161539622411120900403, 8.029118439695131344563334072106, 8.192150902356345798237889214580
