L(s) = 1 | + 1.41·2-s + 2.00·4-s + (2.12 − 0.707i)5-s + (−1 − i)7-s + 2.82·8-s + (3 − 1.00i)10-s − 4.24·11-s + (2 − 2i)13-s + (−1.41 − 1.41i)14-s + 4.00·16-s + (−2.82 + 2.82i)17-s + 6i·19-s + (4.24 − 1.41i)20-s − 6·22-s + (−1.41 + 1.41i)23-s + ⋯ |
L(s) = 1 | + 1.00·2-s + 1.00·4-s + (0.948 − 0.316i)5-s + (−0.377 − 0.377i)7-s + 1.00·8-s + (0.948 − 0.316i)10-s − 1.27·11-s + (0.554 − 0.554i)13-s + (−0.377 − 0.377i)14-s + 1.00·16-s + (−0.685 + 0.685i)17-s + 1.37i·19-s + (0.948 − 0.316i)20-s − 1.27·22-s + (−0.294 + 0.294i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.59632 - 0.302299i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.59632 - 0.302299i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.12 + 0.707i)T \) |
good | 7 | \( 1 + (1 + i)T + 7iT^{2} \) |
| 11 | \( 1 + 4.24T + 11T^{2} \) |
| 13 | \( 1 + (-2 + 2i)T - 13iT^{2} \) |
| 17 | \( 1 + (2.82 - 2.82i)T - 17iT^{2} \) |
| 19 | \( 1 - 6iT - 19T^{2} \) |
| 23 | \( 1 + (1.41 - 1.41i)T - 23iT^{2} \) |
| 29 | \( 1 + 4.24T + 29T^{2} \) |
| 31 | \( 1 + 6iT - 31T^{2} \) |
| 37 | \( 1 + (-8 - 8i)T + 37iT^{2} \) |
| 41 | \( 1 + 8.48T + 41T^{2} \) |
| 43 | \( 1 + (2 + 2i)T + 43iT^{2} \) |
| 47 | \( 1 + (-1.41 - 1.41i)T + 47iT^{2} \) |
| 53 | \( 1 + (1.41 - 1.41i)T - 53iT^{2} \) |
| 59 | \( 1 - 9.89iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + (8 - 8i)T - 67iT^{2} \) |
| 71 | \( 1 + 14.1iT - 71T^{2} \) |
| 73 | \( 1 + (5 + 5i)T + 73iT^{2} \) |
| 79 | \( 1 + 14T + 79T^{2} \) |
| 83 | \( 1 + (1.41 + 1.41i)T + 83iT^{2} \) |
| 89 | \( 1 + 2.82iT - 89T^{2} \) |
| 97 | \( 1 + (-7 + 7i)T - 97iT^{2} \) |
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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
−11.50250643588539637774368192733, −10.36294011668558167203366969043, −10.06343016216194269652216815726, −8.445581395378464207414314083294, −7.50660653904252107201172722963, −6.14017723247317503793466249454, −5.69262108109527268010254781130, −4.45756545464171030231951399474, −3.18207315922795647082309791098, −1.83701143454412288086434799991,
2.18205287743682698429168201225, 3.04493450695017943954254354544, 4.67799044478667742882647035876, 5.56558165605488284856461681045, 6.49858204576268547681713035397, 7.30484255583741505123025240183, 8.772240381169921177461269079756, 9.797249898890391799893987576444, 10.82162703922935660472460576149, 11.39550085916362110915858272871