| L(s) = 1 | + 1.79·3-s − 2.79·5-s + 0.208·9-s + 5.58·11-s + 4·13-s − 5·15-s + 17-s + 2·19-s − 7.58·23-s + 2.79·25-s − 5.00·27-s + 4.79·31-s + 10·33-s − 3.58·37-s + 7.16·39-s + 5.79·41-s + 1.79·43-s − 0.582·45-s + 7.58·47-s + 1.79·51-s − 5.20·53-s − 15.5·55-s + 3.58·57-s + 2.41·59-s + 2.20·61-s − 11.1·65-s + 11.9·67-s + ⋯ |
| L(s) = 1 | + 1.03·3-s − 1.24·5-s + 0.0695·9-s + 1.68·11-s + 1.10·13-s − 1.29·15-s + 0.242·17-s + 0.458·19-s − 1.58·23-s + 0.558·25-s − 0.962·27-s + 0.860·31-s + 1.74·33-s − 0.588·37-s + 1.14·39-s + 0.904·41-s + 0.273·43-s − 0.0868·45-s + 1.10·47-s + 0.250·51-s − 0.715·53-s − 2.10·55-s + 0.474·57-s + 0.314·59-s + 0.282·61-s − 1.38·65-s + 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.370324483\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.370324483\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 1.79T + 3T^{2} \) |
| 5 | \( 1 + 2.79T + 5T^{2} \) |
| 11 | \( 1 - 5.58T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 7.58T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 4.79T + 31T^{2} \) |
| 37 | \( 1 + 3.58T + 37T^{2} \) |
| 41 | \( 1 - 5.79T + 41T^{2} \) |
| 43 | \( 1 - 1.79T + 43T^{2} \) |
| 47 | \( 1 - 7.58T + 47T^{2} \) |
| 53 | \( 1 + 5.20T + 53T^{2} \) |
| 59 | \( 1 - 2.41T + 59T^{2} \) |
| 61 | \( 1 - 2.20T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 - 13.5T + 71T^{2} \) |
| 73 | \( 1 - 10.2T + 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + 0.417T + 83T^{2} \) |
| 89 | \( 1 - 7.16T + 89T^{2} \) |
| 97 | \( 1 - 5.20T + 97T^{2} \) |
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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
−8.392727398063302631771492985602, −8.157185473535855845248277709978, −7.33373535512831631070207449318, −6.48498370050946151082146349990, −5.73186334231391871219904263621, −4.32185421330321881091255475444, −3.79037105057957972672210184380, −3.36242987153296577280979426608, −2.10687829164353309312528916744, −0.903972632988435692391668117711,
0.903972632988435692391668117711, 2.10687829164353309312528916744, 3.36242987153296577280979426608, 3.79037105057957972672210184380, 4.32185421330321881091255475444, 5.73186334231391871219904263621, 6.48498370050946151082146349990, 7.33373535512831631070207449318, 8.157185473535855845248277709978, 8.392727398063302631771492985602
