L(s) = 1 | + (−0.750 + 1.19i)2-s + (−0.707 − 0.707i)3-s + (−0.874 − 1.79i)4-s + (1.07 + 1.95i)5-s + (1.37 − 0.317i)6-s − 1.22·7-s + (2.81 + 0.302i)8-s + 1.00i·9-s + (−3.15 − 0.178i)10-s + (1.38 + 1.38i)11-s + (−0.654 + 1.89i)12-s + (2.12 + 2.12i)13-s + (0.916 − 1.46i)14-s + (0.623 − 2.14i)15-s + (−2.47 + 3.14i)16-s + 6.00i·17-s + ⋯ |
L(s) = 1 | + (−0.530 + 0.847i)2-s + (−0.408 − 0.408i)3-s + (−0.437 − 0.899i)4-s + (0.481 + 0.876i)5-s + (0.562 − 0.129i)6-s − 0.461·7-s + (0.994 + 0.106i)8-s + 0.333i·9-s + (−0.998 − 0.0565i)10-s + (0.416 + 0.416i)11-s + (−0.188 + 0.545i)12-s + (0.588 + 0.588i)13-s + (0.244 − 0.391i)14-s + (0.161 − 0.554i)15-s + (−0.618 + 0.786i)16-s + 1.45i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.227 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.227 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.494402 + 0.622986i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.494402 + 0.622986i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.750 - 1.19i)T \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (-1.07 - 1.95i)T \) |
good | 7 | \( 1 + 1.22T + 7T^{2} \) |
| 11 | \( 1 + (-1.38 - 1.38i)T + 11iT^{2} \) |
| 13 | \( 1 + (-2.12 - 2.12i)T + 13iT^{2} \) |
| 17 | \( 1 - 6.00iT - 17T^{2} \) |
| 19 | \( 1 + (3.06 - 3.06i)T - 19iT^{2} \) |
| 23 | \( 1 - 2.90T + 23T^{2} \) |
| 29 | \( 1 + (-3.18 + 3.18i)T - 29iT^{2} \) |
| 31 | \( 1 + 3.88T + 31T^{2} \) |
| 37 | \( 1 + (-2.44 + 2.44i)T - 37iT^{2} \) |
| 41 | \( 1 - 2.38iT - 41T^{2} \) |
| 43 | \( 1 + (-9.00 + 9.00i)T - 43iT^{2} \) |
| 47 | \( 1 - 0.586iT - 47T^{2} \) |
| 53 | \( 1 + (2.36 - 2.36i)T - 53iT^{2} \) |
| 59 | \( 1 + (-8.43 - 8.43i)T + 59iT^{2} \) |
| 61 | \( 1 + (-9.98 + 9.98i)T - 61iT^{2} \) |
| 67 | \( 1 + (3.82 + 3.82i)T + 67iT^{2} \) |
| 71 | \( 1 + 11.5iT - 71T^{2} \) |
| 73 | \( 1 - 1.31T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 + (2.91 + 2.91i)T + 83iT^{2} \) |
| 89 | \( 1 - 9.58iT - 89T^{2} \) |
| 97 | \( 1 + 9.45iT - 97T^{2} \) |
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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
−12.55186031873485233147402503066, −11.11178501201144898058097032896, −10.42836387488373034508990707379, −9.487278649906544689897875933149, −8.393722485981994868473579765092, −7.19437639530769826113861904896, −6.39702738360576554895217837104, −5.80874842624689226435455719634, −4.04929224825462626145114113248, −1.82615547323473480906988739677,
0.865096743341993992647880739877, 2.89120050012833134741477467682, 4.32279771300566024875871651815, 5.42978436492462153882613872718, 6.85438306526993844799951769115, 8.397532805968769230872711803165, 9.170342999966755209124026818838, 9.850704328408175581080899072159, 10.94042883475728588895253290375, 11.65563865328960228983654612657