L(s) = 1 | + (0.127 + 0.220i)2-s + (0.787 − 1.36i)3-s + (0.967 − 1.67i)4-s + (−0.259 − 0.448i)5-s + 0.401·6-s + (−1.92 + 1.81i)7-s + 1.00·8-s + (0.259 + 0.449i)9-s + (0.0659 − 0.114i)10-s + (0.648 − 1.12i)11-s + (−1.52 − 2.63i)12-s + (−0.645 − 0.193i)14-s − 0.815·15-s + (−1.80 − 3.13i)16-s + (1.85 − 3.21i)17-s + (−0.0660 + 0.114i)18-s + ⋯ |
L(s) = 1 | + (0.0900 + 0.156i)2-s + (0.454 − 0.787i)3-s + (0.483 − 0.837i)4-s + (−0.115 − 0.200i)5-s + 0.163·6-s + (−0.727 + 0.686i)7-s + 0.354·8-s + (0.0864 + 0.149i)9-s + (0.0208 − 0.0361i)10-s + (0.195 − 0.338i)11-s + (−0.439 − 0.762i)12-s + (−0.172 − 0.0516i)14-s − 0.210·15-s + (−0.451 − 0.782i)16-s + (0.449 − 0.779i)17-s + (−0.0155 + 0.0269i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.346 + 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.346 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.916936789\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.916936789\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (1.92 - 1.81i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.127 - 0.220i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.787 + 1.36i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (0.259 + 0.448i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.648 + 1.12i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.85 + 3.21i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.20 + 3.82i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.50 + 6.06i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 1.85T + 29T^{2} \) |
| 31 | \( 1 + (0.299 - 0.518i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.39 + 4.15i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 0.942T + 41T^{2} \) |
| 43 | \( 1 + 0.546T + 43T^{2} \) |
| 47 | \( 1 + (-4.77 - 8.27i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (5.21 - 9.03i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.32 + 10.9i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.55 - 6.16i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.19 + 5.53i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 14.7T + 71T^{2} \) |
| 73 | \( 1 + (6.47 - 11.2i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.63 - 11.4i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 + (-3.86 - 6.69i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 4.09T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.402484395666465550063727215482, −8.702097589949812311870835275844, −7.79186400669831607250927239641, −6.86474210256238603244245504220, −6.36099282983589020833689368145, −5.42818738698285329339701650538, −4.43914404596030244190848226446, −2.81505109470981564407221542358, −2.19014258350800569923959201225, −0.74391984905280028931741055535,
1.77921763543481906850031328176, 3.37340219375142788197023162738, 3.57115594436206895618381078807, 4.42735473782474756486356269987, 5.91174316650987471484144442318, 6.86393890796416695242043765935, 7.52714878813313602308554570962, 8.402749756453847436966497172866, 9.293469488323452480587094794192, 10.14419289816589613905421958031