L(s) = 1 | + (−2.12 − 2.12i)5-s − 7-s + (4.03 − 4.03i)11-s + (−4.91 − 4.91i)13-s + 4.76i·17-s + (2.21 − 2.21i)19-s − 6.91i·23-s + 3.99i·25-s + (−2.61 + 2.61i)29-s − 0.712i·31-s + (2.12 + 2.12i)35-s + (5.73 − 5.73i)37-s + 3.59·41-s + (−3.36 − 3.36i)43-s + 6.04·47-s + ⋯ |
L(s) = 1 | + (−0.948 − 0.948i)5-s − 0.377·7-s + (1.21 − 1.21i)11-s + (−1.36 − 1.36i)13-s + 1.15i·17-s + (0.508 − 0.508i)19-s − 1.44i·23-s + 0.798i·25-s + (−0.485 + 0.485i)29-s − 0.127i·31-s + (0.358 + 0.358i)35-s + (0.943 − 0.943i)37-s + 0.561·41-s + (−0.512 − 0.512i)43-s + 0.881·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 - 0.268i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.963 - 0.268i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7625890543\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7625890543\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
good | 5 | \( 1 + (2.12 + 2.12i)T + 5iT^{2} \) |
| 11 | \( 1 + (-4.03 + 4.03i)T - 11iT^{2} \) |
| 13 | \( 1 + (4.91 + 4.91i)T + 13iT^{2} \) |
| 17 | \( 1 - 4.76iT - 17T^{2} \) |
| 19 | \( 1 + (-2.21 + 2.21i)T - 19iT^{2} \) |
| 23 | \( 1 + 6.91iT - 23T^{2} \) |
| 29 | \( 1 + (2.61 - 2.61i)T - 29iT^{2} \) |
| 31 | \( 1 + 0.712iT - 31T^{2} \) |
| 37 | \( 1 + (-5.73 + 5.73i)T - 37iT^{2} \) |
| 41 | \( 1 - 3.59T + 41T^{2} \) |
| 43 | \( 1 + (3.36 + 3.36i)T + 43iT^{2} \) |
| 47 | \( 1 - 6.04T + 47T^{2} \) |
| 53 | \( 1 + (4.53 + 4.53i)T + 53iT^{2} \) |
| 59 | \( 1 + (4.82 - 4.82i)T - 59iT^{2} \) |
| 61 | \( 1 + (6.72 + 6.72i)T + 61iT^{2} \) |
| 67 | \( 1 + (-3.87 + 3.87i)T - 67iT^{2} \) |
| 71 | \( 1 - 14.7iT - 71T^{2} \) |
| 73 | \( 1 - 4.61iT - 73T^{2} \) |
| 79 | \( 1 + 7.43iT - 79T^{2} \) |
| 83 | \( 1 + (-2.44 - 2.44i)T + 83iT^{2} \) |
| 89 | \( 1 + 4.23T + 89T^{2} \) |
| 97 | \( 1 + 6.76T + 97T^{2} \) |
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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
−8.070129549440051159040779875199, −7.51286107628832465524260563647, −6.53907968315827727073821115239, −5.78979825127022347412502225941, −5.00155470364801613580376242431, −4.15099663769928546936367965804, −3.52005177339161516400664256500, −2.56892768361251018009971368051, −1.00151571450968916571004246979, −0.26311671685749726060416952298,
1.55915545094469068721942793025, 2.61347804514321725523833577977, 3.48091923205693002021652903953, 4.28043822475345523633346428158, 4.84551634052185757409791542866, 6.12486700734412348352071524954, 6.86747176515553714895168631781, 7.42485450343007682940283563298, 7.62764028693251095074927713015, 9.109128032016983132233520648388