| L(s) = 1 | + 1.79·3-s − 4.79·7-s + 0.208·9-s + 11-s + 13-s + 3.79·17-s + 2.58·19-s − 8.58·21-s + 0.791·23-s − 5.00·27-s + 2.20·29-s − 0.582·31-s + 1.79·33-s − 6.58·37-s + 1.79·39-s − 10.5·41-s − 10·43-s − 10.5·47-s + 15.9·49-s + 6.79·51-s + 2.37·53-s + 4.62·57-s + 1.41·59-s + 8.79·61-s − 0.999·63-s − 4·67-s + 1.41·69-s + ⋯ |
| L(s) = 1 | + 1.03·3-s − 1.81·7-s + 0.0695·9-s + 0.301·11-s + 0.277·13-s + 0.919·17-s + 0.592·19-s − 1.87·21-s + 0.164·23-s − 0.962·27-s + 0.410·29-s − 0.104·31-s + 0.311·33-s − 1.08·37-s + 0.286·39-s − 1.65·41-s − 1.52·43-s − 1.54·47-s + 2.27·49-s + 0.950·51-s + 0.326·53-s + 0.612·57-s + 0.184·59-s + 1.12·61-s − 0.125·63-s − 0.488·67-s + 0.170·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 1.79T + 3T^{2} \) |
| 7 | \( 1 + 4.79T + 7T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 - 3.79T + 17T^{2} \) |
| 19 | \( 1 - 2.58T + 19T^{2} \) |
| 23 | \( 1 - 0.791T + 23T^{2} \) |
| 29 | \( 1 - 2.20T + 29T^{2} \) |
| 31 | \( 1 + 0.582T + 31T^{2} \) |
| 37 | \( 1 + 6.58T + 37T^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 + 10T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 - 2.37T + 53T^{2} \) |
| 59 | \( 1 - 1.41T + 59T^{2} \) |
| 61 | \( 1 - 8.79T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 16.7T + 71T^{2} \) |
| 73 | \( 1 - 3.20T + 73T^{2} \) |
| 79 | \( 1 + 16.5T + 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 - 3.79T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
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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
−8.187366243876962278631842569178, −7.19864335444048231302553322258, −6.67995616845854747281553753748, −5.89396895749864864133880477882, −5.07420429000274987707134614306, −3.69930839427493440531947344301, −3.36831715496259971684205532511, −2.76852539533625083743296053621, −1.52142825204025655523457578620, 0,
1.52142825204025655523457578620, 2.76852539533625083743296053621, 3.36831715496259971684205532511, 3.69930839427493440531947344301, 5.07420429000274987707134614306, 5.89396895749864864133880477882, 6.67995616845854747281553753748, 7.19864335444048231302553322258, 8.187366243876962278631842569178
