L(s) = 1 | + (−0.707 + 1.22i)2-s + (−0.707 + 0.707i)3-s + (−0.999 − 1.73i)4-s + (−0.366 − 1.36i)6-s + (3.15 + 3.15i)7-s + 2.82·8-s − 1.00i·9-s + 2i·11-s + (1.93 + 0.517i)12-s + (−1.60 − 1.60i)13-s + (−6.09 + 1.63i)14-s + (−2.00 + 3.46i)16-s + (−4.24 + 4.24i)17-s + (1.22 + 0.707i)18-s + 3.19·19-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.866i)2-s + (−0.408 + 0.408i)3-s + (−0.499 − 0.866i)4-s + (−0.149 − 0.557i)6-s + (1.19 + 1.19i)7-s + 0.999·8-s − 0.333i·9-s + 0.603i·11-s + (0.557 + 0.149i)12-s + (−0.444 − 0.444i)13-s + (−1.62 + 0.436i)14-s + (−0.500 + 0.866i)16-s + (−1.02 + 1.02i)17-s + (0.288 + 0.166i)18-s + 0.733·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.292321 + 0.782911i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.292321 + 0.782911i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 1.22i)T \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-3.15 - 3.15i)T + 7iT^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 13 | \( 1 + (1.60 + 1.60i)T + 13iT^{2} \) |
| 17 | \( 1 + (4.24 - 4.24i)T - 17iT^{2} \) |
| 19 | \( 1 - 3.19T + 19T^{2} \) |
| 23 | \( 1 + (5.27 - 5.27i)T - 23iT^{2} \) |
| 29 | \( 1 + 0.535iT - 29T^{2} \) |
| 31 | \( 1 - 3.73iT - 31T^{2} \) |
| 37 | \( 1 + (-7.72 + 7.72i)T - 37iT^{2} \) |
| 41 | \( 1 - 1.46T + 41T^{2} \) |
| 43 | \( 1 + (3.15 - 3.15i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.656 + 0.656i)T + 47iT^{2} \) |
| 53 | \( 1 + (-3.86 - 3.86i)T + 53iT^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 - 3T + 61T^{2} \) |
| 67 | \( 1 + (3.91 + 3.91i)T + 67iT^{2} \) |
| 71 | \( 1 + 2.53iT - 71T^{2} \) |
| 73 | \( 1 + (-2.82 - 2.82i)T + 73iT^{2} \) |
| 79 | \( 1 + 7.46T + 79T^{2} \) |
| 83 | \( 1 + (-9.14 + 9.14i)T - 83iT^{2} \) |
| 89 | \( 1 + 6.92iT - 89T^{2} \) |
| 97 | \( 1 + (-6.88 + 6.88i)T - 97iT^{2} \) |
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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
−11.89751070121739482745073009571, −11.08480740080487544586750004581, −10.05489461205211819124051959137, −9.170373605603036857191866796821, −8.283902235055304522476029838913, −7.42174811574205382612199055818, −6.01885798963347383938639929712, −5.31778834999947514289019944709, −4.34424633014078671231037675409, −1.92753299745180275473370532683,
0.803625331949987604251320135723, 2.31723318336624174704414250190, 4.10618377130692372675998514176, 4.96095121256671559131796653808, 6.77830035967490263310000599077, 7.69430605405157103261854784641, 8.460345151013375483154990769837, 9.732161559996905788141181637893, 10.62741591981979422901082805113, 11.48665803916868616207868375247