L(s) = 1 | + (0.707 − 0.707i)2-s + 0.999i·4-s − 2.23i·5-s + (0.581 + 2.58i)7-s + (2.12 + 2.12i)8-s + (−1.58 − 1.58i)10-s + 4.24·11-s + (3.16 − 3.16i)13-s + (2.23 + 1.41i)14-s + 1.00·16-s + (−2.23 − 2.23i)17-s − 3.16·19-s + 2.23·20-s + (3 − 3i)22-s + (1.41 + 1.41i)23-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + 0.499i·4-s − 0.999i·5-s + (0.219 + 0.975i)7-s + (0.750 + 0.750i)8-s + (−0.500 − 0.500i)10-s + 1.27·11-s + (0.877 − 0.877i)13-s + (0.597 + 0.377i)14-s + 0.250·16-s + (−0.542 − 0.542i)17-s − 0.725·19-s + 0.499·20-s + (0.639 − 0.639i)22-s + (0.294 + 0.294i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.80991 - 0.303346i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.80991 - 0.303346i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 7 | \( 1 + (-0.581 - 2.58i)T \) |
good | 2 | \( 1 + (-0.707 + 0.707i)T - 2iT^{2} \) |
| 11 | \( 1 - 4.24T + 11T^{2} \) |
| 13 | \( 1 + (-3.16 + 3.16i)T - 13iT^{2} \) |
| 17 | \( 1 + (2.23 + 2.23i)T + 17iT^{2} \) |
| 19 | \( 1 + 3.16T + 19T^{2} \) |
| 23 | \( 1 + (-1.41 - 1.41i)T + 23iT^{2} \) |
| 29 | \( 1 + 5.65iT - 29T^{2} \) |
| 31 | \( 1 - 9.48iT - 31T^{2} \) |
| 37 | \( 1 + (-1 + i)T - 37iT^{2} \) |
| 41 | \( 1 - 8.94iT - 41T^{2} \) |
| 43 | \( 1 + (2 + 2i)T + 43iT^{2} \) |
| 47 | \( 1 + (8.94 + 8.94i)T + 47iT^{2} \) |
| 53 | \( 1 + (7.07 + 7.07i)T + 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + (-4 + 4i)T - 67iT^{2} \) |
| 71 | \( 1 + 4.24T + 71T^{2} \) |
| 73 | \( 1 + (-3.16 + 3.16i)T - 73iT^{2} \) |
| 79 | \( 1 - 12iT - 79T^{2} \) |
| 83 | \( 1 + (4.47 - 4.47i)T - 83iT^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 + (-3.16 - 3.16i)T + 97iT^{2} \) |
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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
−11.71100454988700413293284909885, −11.16061699159095404357405107709, −9.625504257449301325720916088863, −8.591975975190096521598035399474, −8.237943200128175041469385331301, −6.59558931114353958790343078182, −5.35455342629090684011593786563, −4.41941528038749870043094090800, −3.25514935920689086068508982094, −1.71249064078519069153370415083,
1.61457414064169123493372154367, 3.78214199514477582146663832276, 4.43580422544033213529847670309, 6.21707355653068951582457757729, 6.55428585255748950682630328750, 7.49113339480393959591834253839, 8.938417816302828828159955236850, 9.992252391008034748000131049856, 10.93917802570975408733316732746, 11.34680894567809506893574034708