L(s) = 1 | + (−2 + 2i)3-s + (−1 − 2i)5-s + (2 − 2i)7-s − 5i·9-s + (1 + i)13-s + (6 + 2i)15-s + (−5 − 5i)17-s + 4i·19-s + 8i·21-s + (2 + 2i)23-s + (−3 + 4i)25-s + (4 + 4i)27-s − 4·29-s + 4i·31-s + (−6 − 2i)35-s + ⋯ |
L(s) = 1 | + (−1.15 + 1.15i)3-s + (−0.447 − 0.894i)5-s + (0.755 − 0.755i)7-s − 1.66i·9-s + (0.277 + 0.277i)13-s + (1.54 + 0.516i)15-s + (−1.21 − 1.21i)17-s + 0.917i·19-s + 1.74i·21-s + (0.417 + 0.417i)23-s + (−0.600 + 0.800i)25-s + (0.769 + 0.769i)27-s − 0.742·29-s + 0.718i·31-s + (−1.01 − 0.338i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (1 + 2i)T \) |
good | 3 | \( 1 + (2 - 2i)T - 3iT^{2} \) |
| 7 | \( 1 + (-2 + 2i)T - 7iT^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + (-1 - i)T + 13iT^{2} \) |
| 17 | \( 1 + (5 + 5i)T + 17iT^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + (-2 - 2i)T + 23iT^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 - 4iT - 31T^{2} \) |
| 37 | \( 1 + (1 - i)T - 37iT^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + (6 - 6i)T - 43iT^{2} \) |
| 47 | \( 1 + (-2 + 2i)T - 47iT^{2} \) |
| 53 | \( 1 + (7 + 7i)T + 53iT^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 - 4iT - 61T^{2} \) |
| 67 | \( 1 + (10 + 10i)T + 67iT^{2} \) |
| 71 | \( 1 - 12iT - 71T^{2} \) |
| 73 | \( 1 + (-3 + 3i)T - 73iT^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 + (-2 + 2i)T - 83iT^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + (3 + 3i)T + 97iT^{2} \) |
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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
−9.415557178256880585550460888635, −8.660546190363913775323305636911, −7.67331723662298675710409781987, −6.73288724302142517601136206848, −5.61972183963406343506377824771, −4.83519273846649149134888904510, −4.43703331194059424384529753932, −3.53645138090801609262295088117, −1.41325110075920260156811245777, 0,
1.67958412281817577727598520158, 2.61411229920281208829107491547, 4.15493522740446721861832867314, 5.24336032617014364295283870609, 6.08344209950035403578235608889, 6.67051044824535233303528726229, 7.44038374413121153569191694836, 8.210449272923174687551421527910, 9.029761147216760046896227887824