L(s) = 1 | + (−0.965 + 1.67i)5-s + (−2.53 + 0.768i)7-s + (1.10 − 0.639i)11-s + (2.52 + 1.45i)13-s + 0.475·17-s + 6.10i·19-s + (−6.51 − 3.75i)23-s + (0.635 + 1.09i)25-s + (−3.76 + 2.17i)29-s + (4.21 + 2.43i)31-s + (1.15 − 4.97i)35-s + 1.76·37-s + (1.16 − 2.01i)41-s + (−4.63 − 8.03i)43-s + (−4.00 − 6.93i)47-s + ⋯ |
L(s) = 1 | + (−0.431 + 0.747i)5-s + (−0.956 + 0.290i)7-s + (0.333 − 0.192i)11-s + (0.701 + 0.404i)13-s + 0.115·17-s + 1.40i·19-s + (−1.35 − 0.783i)23-s + (0.127 + 0.219i)25-s + (−0.699 + 0.403i)29-s + (0.757 + 0.437i)31-s + (0.195 − 0.841i)35-s + 0.289·37-s + (0.181 − 0.314i)41-s + (−0.707 − 1.22i)43-s + (−0.584 − 1.01i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3541190968\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3541190968\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.53 - 0.768i)T \) |
good | 5 | \( 1 + (0.965 - 1.67i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.10 + 0.639i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.52 - 1.45i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 0.475T + 17T^{2} \) |
| 19 | \( 1 - 6.10iT - 19T^{2} \) |
| 23 | \( 1 + (6.51 + 3.75i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.76 - 2.17i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.21 - 2.43i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 1.76T + 37T^{2} \) |
| 41 | \( 1 + (-1.16 + 2.01i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4.63 + 8.03i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.00 + 6.93i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 10.3iT - 53T^{2} \) |
| 59 | \( 1 + (1.74 - 3.01i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.26 - 2.46i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.602 - 1.04i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 11.2iT - 71T^{2} \) |
| 73 | \( 1 + 1.84iT - 73T^{2} \) |
| 79 | \( 1 + (8.54 + 14.8i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.225 + 0.390i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 11.2T + 89T^{2} \) |
| 97 | \( 1 + (9.22 - 5.32i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.021656947147791641498907271654, −8.454101401348492280671904486430, −7.56918935875651432024240294400, −6.78839024789524375327327787652, −6.18463883771323112871696826130, −5.56282954470542442303382474632, −4.13034278609734792903895711956, −3.62795536618997489869159936671, −2.80461307099729100157731444321, −1.60251692013669308152128522105,
0.11784998923456005942184002074, 1.23286777496838752586433457066, 2.65405546531850829615085390475, 3.62954990616623265792466330630, 4.30200046871467052267629743048, 5.16165671279759506752019257013, 6.20344684555394277780632199524, 6.63929538102466458348352402240, 7.79508616050172162358963482389, 8.159094197226960462418808769671