L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s + (−2 − 2i)7-s + (−0.707 − 0.707i)8-s − 2.82i·11-s + (3 − 3i)13-s − 2.82·14-s − 1.00·16-s + (−2.82 + 2.82i)17-s − 8i·19-s + (−2.00 − 2.00i)22-s + (2.82 + 2.82i)23-s − 4.24i·26-s + (−2.00 + 2.00i)28-s − 1.41·29-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s − 0.500i·4-s + (−0.755 − 0.755i)7-s + (−0.250 − 0.250i)8-s − 0.852i·11-s + (0.832 − 0.832i)13-s − 0.755·14-s − 0.250·16-s + (−0.685 + 0.685i)17-s − 1.83i·19-s + (−0.426 − 0.426i)22-s + (0.589 + 0.589i)23-s − 0.832i·26-s + (−0.377 + 0.377i)28-s − 0.262·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.374 + 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.867362 - 1.28547i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.867362 - 1.28547i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (2 + 2i)T + 7iT^{2} \) |
| 11 | \( 1 + 2.82iT - 11T^{2} \) |
| 13 | \( 1 + (-3 + 3i)T - 13iT^{2} \) |
| 17 | \( 1 + (2.82 - 2.82i)T - 17iT^{2} \) |
| 19 | \( 1 + 8iT - 19T^{2} \) |
| 23 | \( 1 + (-2.82 - 2.82i)T + 23iT^{2} \) |
| 29 | \( 1 + 1.41T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + (-3 - 3i)T + 37iT^{2} \) |
| 41 | \( 1 - 9.89iT - 41T^{2} \) |
| 43 | \( 1 - 43iT^{2} \) |
| 47 | \( 1 - 47iT^{2} \) |
| 53 | \( 1 + (2.82 + 2.82i)T + 53iT^{2} \) |
| 59 | \( 1 - 2.82T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + (-4 - 4i)T + 67iT^{2} \) |
| 71 | \( 1 + 5.65iT - 71T^{2} \) |
| 73 | \( 1 + (1 - i)T - 73iT^{2} \) |
| 79 | \( 1 - 12iT - 79T^{2} \) |
| 83 | \( 1 + (-8.48 - 8.48i)T + 83iT^{2} \) |
| 89 | \( 1 - 9.89T + 89T^{2} \) |
| 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
−11.01741655733978773939803375836, −10.11738057473492511181494849375, −9.147488110910571738775811353048, −8.171731286314298625227412212456, −6.82742607492512734732687657182, −6.11648095784285836951446656422, −4.88818156063063293816677009933, −3.70381164585926326203670811295, −2.85474579952189291429070302072, −0.852917136872713109943982742994,
2.19325509155823993485825402202, 3.58731091892657311913817417962, 4.64264949554590850888125351361, 5.87133931367435256965919071251, 6.54964679764010599112648096555, 7.53476956610735752502188404998, 8.716928959837207772650199635794, 9.373469719333645438184249215146, 10.44930049945456477171009077035, 11.64391103104480131664110395668