L(s) = 1 | + 1.73i·2-s + 3-s − 0.999·4-s + 1.73i·6-s − 3.46i·7-s + 1.73i·8-s + 9-s + 3.46i·11-s − 0.999·12-s + (1 + 3.46i)13-s + 5.99·14-s − 5·16-s + 6·17-s + 1.73i·18-s − 3.46i·19-s + ⋯ |
L(s) = 1 | + 1.22i·2-s + 0.577·3-s − 0.499·4-s + 0.707i·6-s − 1.30i·7-s + 0.612i·8-s + 0.333·9-s + 1.04i·11-s − 0.288·12-s + (0.277 + 0.960i)13-s + 1.60·14-s − 1.25·16-s + 1.45·17-s + 0.408i·18-s − 0.794i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.28354 + 1.70647i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.28354 + 1.70647i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-1 - 3.46i)T \) |
good | 2 | \( 1 - 1.73iT - 2T^{2} \) |
| 7 | \( 1 + 3.46iT - 7T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 3.46iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 3.46iT - 31T^{2} \) |
| 37 | \( 1 - 6.92iT - 37T^{2} \) |
| 41 | \( 1 - 6.92iT - 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 3.46iT - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 10.3iT - 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 10.3iT - 67T^{2} \) |
| 71 | \( 1 + 3.46iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 3.46iT - 83T^{2} \) |
| 89 | \( 1 - 6.92iT - 89T^{2} \) |
| 97 | \( 1 + 13.8iT - 97T^{2} \) |
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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
−10.00467916272955544818941607342, −9.316193324389101243668353090925, −8.234756452490219312964017805000, −7.63570519375726352553717664062, −6.93548174906554347930325070034, −6.38538022154967726230096877373, −4.93266163837314642554371749871, −4.35179957296806594844870933078, −3.04262210108616053212904394019, −1.49808899212528247292178915742,
1.05953510937097008657874850035, 2.40868098729543463584390781909, 3.10809022980903272446505129450, 3.88265266079115475206211285074, 5.47194297151417843974516966826, 6.05462746589581197484607584344, 7.50631960058184002653580088982, 8.395421513078788980515219485775, 9.012525604542732739739282197235, 9.950883509439329918928548855049