| L(s) = 1 | + 0.409·2-s + 3-s − 1.83·4-s + 4.13·5-s + 0.409·6-s + 5.18·7-s − 1.56·8-s + 9-s + 1.69·10-s + 11-s − 1.83·12-s + 2.12·14-s + 4.13·15-s + 3.02·16-s − 0.488·17-s + 0.409·18-s − 0.446·19-s − 7.58·20-s + 5.18·21-s + 0.409·22-s + 5.50·23-s − 1.56·24-s + 12.1·25-s + 27-s − 9.49·28-s − 6.58·29-s + 1.69·30-s + ⋯ |
| L(s) = 1 | + 0.289·2-s + 0.577·3-s − 0.916·4-s + 1.85·5-s + 0.167·6-s + 1.95·7-s − 0.554·8-s + 0.333·9-s + 0.535·10-s + 0.301·11-s − 0.529·12-s + 0.566·14-s + 1.06·15-s + 0.755·16-s − 0.118·17-s + 0.0964·18-s − 0.102·19-s − 1.69·20-s + 1.13·21-s + 0.0872·22-s + 1.14·23-s − 0.320·24-s + 2.42·25-s + 0.192·27-s − 1.79·28-s − 1.22·29-s + 0.309·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.487718644\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.487718644\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.409T + 2T^{2} \) |
| 5 | \( 1 - 4.13T + 5T^{2} \) |
| 7 | \( 1 - 5.18T + 7T^{2} \) |
| 17 | \( 1 + 0.488T + 17T^{2} \) |
| 19 | \( 1 + 0.446T + 19T^{2} \) |
| 23 | \( 1 - 5.50T + 23T^{2} \) |
| 29 | \( 1 + 6.58T + 29T^{2} \) |
| 31 | \( 1 - 1.52T + 31T^{2} \) |
| 37 | \( 1 + 7.87T + 37T^{2} \) |
| 41 | \( 1 + 8.78T + 41T^{2} \) |
| 43 | \( 1 + 4.57T + 43T^{2} \) |
| 47 | \( 1 - 5.42T + 47T^{2} \) |
| 53 | \( 1 - 8.66T + 53T^{2} \) |
| 59 | \( 1 - 5.60T + 59T^{2} \) |
| 61 | \( 1 + 0.855T + 61T^{2} \) |
| 67 | \( 1 + 3.87T + 67T^{2} \) |
| 71 | \( 1 + 8.49T + 71T^{2} \) |
| 73 | \( 1 - 5.95T + 73T^{2} \) |
| 79 | \( 1 + 8.91T + 79T^{2} \) |
| 83 | \( 1 - 0.457T + 83T^{2} \) |
| 89 | \( 1 + 4.68T + 89T^{2} \) |
| 97 | \( 1 + 2.06T + 97T^{2} \) |
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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
−8.536663884805517740928998526721, −7.47421072095130205249139632218, −6.71134417585377175067223699260, −5.60428471390975845356544084083, −5.26484402839944358196259785404, −4.68744617595010562436299196987, −3.78877222794417874814822120777, −2.67266729612665097313494257753, −1.79222977389561937264660185332, −1.23943338362542005490732876099,
1.23943338362542005490732876099, 1.79222977389561937264660185332, 2.67266729612665097313494257753, 3.78877222794417874814822120777, 4.68744617595010562436299196987, 5.26484402839944358196259785404, 5.60428471390975845356544084083, 6.71134417585377175067223699260, 7.47421072095130205249139632218, 8.536663884805517740928998526721
