| L(s) = 1 | + 1.73i·2-s − 0.999·4-s − 2i·7-s + 1.73i·8-s + 11-s − 1.46i·13-s + 3.46·14-s − 5·16-s + 1.46·19-s + 1.73i·22-s + 6.92i·23-s + 2.53·26-s + 1.99i·28-s + 3.46·29-s + 2.92·31-s − 5.19i·32-s + ⋯ |
| L(s) = 1 | + 1.22i·2-s − 0.499·4-s − 0.755i·7-s + 0.612i·8-s + 0.301·11-s − 0.406i·13-s + 0.925·14-s − 1.25·16-s + 0.335·19-s + 0.369i·22-s + 1.44i·23-s + 0.497·26-s + 0.377i·28-s + 0.643·29-s + 0.525·31-s − 0.918i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.884876107\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.884876107\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 1.73iT - 2T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 13 | \( 1 + 1.46iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 1.46T + 19T^{2} \) |
| 23 | \( 1 - 6.92iT - 23T^{2} \) |
| 29 | \( 1 - 3.46T + 29T^{2} \) |
| 31 | \( 1 - 2.92T + 31T^{2} \) |
| 37 | \( 1 + 8.92iT - 37T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 - 8.92iT - 43T^{2} \) |
| 47 | \( 1 - 6.92iT - 47T^{2} \) |
| 53 | \( 1 - 12.9iT - 53T^{2} \) |
| 59 | \( 1 - 6.92T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 - 12.3iT - 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 + 15.4iT - 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−9.072222732750881293008469496226, −8.025572296018133859247265850709, −7.62762012153946980199403947133, −6.94840800859392313347691114075, −6.15477963843348821495211188053, −5.49159182937081071445558283649, −4.61623785738444110451720154282, −3.71528161182424908862997565148, −2.54743812765782901692226842624, −1.09206074256356133966149196844,
0.74075878314470059071410465677, 1.99802727104777737060257644841, 2.66057316537643582387278013236, 3.61095917898919363371828962746, 4.47140250956091019019732810514, 5.37314233472007462278590061308, 6.51761442944715617598222559739, 6.92105846479597806481536667285, 8.307844783295276474903696272500, 8.752721436561826994890918868769
