| L(s) = 1 | + 4.99·7-s + 3.99·11-s − 1.54·13-s + 6.99·17-s − 2.25·19-s − 7.79·23-s + 6.16·29-s − 0.543·31-s + 6.25·37-s + 0.195·41-s + 0.0863·43-s + 3.82·47-s + 17.9·49-s − 4.19·53-s + 7.02·59-s − 2.90·61-s − 8.96·67-s + 8.79·71-s + 2.28·73-s + 19.9·77-s − 12.6·79-s + 13.9·83-s − 10.3·89-s − 7.70·91-s − 9.33·97-s + 7.96·101-s + 16.4·103-s + ⋯ |
| L(s) = 1 | + 1.88·7-s + 1.20·11-s − 0.427·13-s + 1.69·17-s − 0.517·19-s − 1.62·23-s + 1.14·29-s − 0.0975·31-s + 1.02·37-s + 0.0305·41-s + 0.0131·43-s + 0.558·47-s + 2.56·49-s − 0.575·53-s + 0.914·59-s − 0.372·61-s − 1.09·67-s + 1.04·71-s + 0.267·73-s + 2.27·77-s − 1.42·79-s + 1.53·83-s − 1.09·89-s − 0.808·91-s − 0.948·97-s + 0.792·101-s + 1.61·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.142513987\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.142513987\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 4.99T + 7T^{2} \) |
| 11 | \( 1 - 3.99T + 11T^{2} \) |
| 13 | \( 1 + 1.54T + 13T^{2} \) |
| 17 | \( 1 - 6.99T + 17T^{2} \) |
| 19 | \( 1 + 2.25T + 19T^{2} \) |
| 23 | \( 1 + 7.79T + 23T^{2} \) |
| 29 | \( 1 - 6.16T + 29T^{2} \) |
| 31 | \( 1 + 0.543T + 31T^{2} \) |
| 37 | \( 1 - 6.25T + 37T^{2} \) |
| 41 | \( 1 - 0.195T + 41T^{2} \) |
| 43 | \( 1 - 0.0863T + 43T^{2} \) |
| 47 | \( 1 - 3.82T + 47T^{2} \) |
| 53 | \( 1 + 4.19T + 53T^{2} \) |
| 59 | \( 1 - 7.02T + 59T^{2} \) |
| 61 | \( 1 + 2.90T + 61T^{2} \) |
| 67 | \( 1 + 8.96T + 67T^{2} \) |
| 71 | \( 1 - 8.79T + 71T^{2} \) |
| 73 | \( 1 - 2.28T + 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 - 13.9T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 + 9.33T + 97T^{2} \) |
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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
−7.84668583910889322969392129230, −7.36867348337608727973841359174, −6.34064509761439188588033920284, −5.74807244281112578111891810364, −4.94324656103658913933673177475, −4.33010210961787817089678907288, −3.70251227430602146089908453671, −2.50281924970209891448838097383, −1.65874169577518538026712957340, −0.968716165381294732876522228342,
0.968716165381294732876522228342, 1.65874169577518538026712957340, 2.50281924970209891448838097383, 3.70251227430602146089908453671, 4.33010210961787817089678907288, 4.94324656103658913933673177475, 5.74807244281112578111891810364, 6.34064509761439188588033920284, 7.36867348337608727973841359174, 7.84668583910889322969392129230
