L(s) = 1 | + (−1.19 − 0.760i)2-s + (0.844 + 1.81i)4-s + (0.611 + 0.611i)7-s + (0.371 − 2.80i)8-s − 5.12i·11-s + (−1.76 − 1.76i)13-s + (−0.264 − 1.19i)14-s + (−2.57 + 3.06i)16-s + (−3.76 + 3.76i)17-s + 1.22·19-s + (−3.89 + 6.11i)22-s + (1.07 − 1.07i)23-s + (0.761 + 3.43i)26-s + (−0.592 + 1.62i)28-s − 0.864i·29-s + ⋯ |
L(s) = 1 | + (−0.843 − 0.537i)2-s + (0.422 + 0.906i)4-s + (0.231 + 0.231i)7-s + (0.131 − 0.991i)8-s − 1.54i·11-s + (−0.488 − 0.488i)13-s + (−0.0706 − 0.319i)14-s + (−0.643 + 0.765i)16-s + (−0.912 + 0.912i)17-s + 0.280·19-s + (−0.831 + 1.30i)22-s + (0.224 − 0.224i)23-s + (0.149 + 0.674i)26-s + (−0.111 + 0.307i)28-s − 0.160i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.549 + 0.835i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.549 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.353454 - 0.655275i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.353454 - 0.655275i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.19 + 0.760i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-0.611 - 0.611i)T + 7iT^{2} \) |
| 11 | \( 1 + 5.12iT - 11T^{2} \) |
| 13 | \( 1 + (1.76 + 1.76i)T + 13iT^{2} \) |
| 17 | \( 1 + (3.76 - 3.76i)T - 17iT^{2} \) |
| 19 | \( 1 - 1.22T + 19T^{2} \) |
| 23 | \( 1 + (-1.07 + 1.07i)T - 23iT^{2} \) |
| 29 | \( 1 + 0.864iT - 29T^{2} \) |
| 31 | \( 1 + 7.81iT - 31T^{2} \) |
| 37 | \( 1 + (-1.76 + 1.76i)T - 37iT^{2} \) |
| 41 | \( 1 + 5.52T + 41T^{2} \) |
| 43 | \( 1 + (-6.20 + 6.20i)T - 43iT^{2} \) |
| 47 | \( 1 + (2.29 + 2.29i)T + 47iT^{2} \) |
| 53 | \( 1 + (2.62 + 2.62i)T + 53iT^{2} \) |
| 59 | \( 1 + 0.528T + 59T^{2} \) |
| 61 | \( 1 - 4.98T + 61T^{2} \) |
| 67 | \( 1 + (6.20 + 6.20i)T + 67iT^{2} \) |
| 71 | \( 1 - 8.10iT - 71T^{2} \) |
| 73 | \( 1 + (-2.25 - 2.25i)T + 73iT^{2} \) |
| 79 | \( 1 + 15.9T + 79T^{2} \) |
| 83 | \( 1 + (-7.95 + 7.95i)T - 83iT^{2} \) |
| 89 | \( 1 + 7.25iT - 89T^{2} \) |
| 97 | \( 1 + (0.793 - 0.793i)T - 97iT^{2} \) |
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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
−9.875377439707781126691829823574, −8.858297783803280304010725530567, −8.395308156256568398056289922351, −7.56206660163326632284949107402, −6.47792282172055872542212608430, −5.57660977334623906491623981323, −4.14562710014348324545270558475, −3.13980175067048337126214502736, −2.05834774117868516366409882247, −0.47205819931704010190737449149,
1.48474782716240054844806024777, 2.63279016692604674905004931292, 4.52865388948965708978292104080, 5.06565017605079107437259201796, 6.45320671270648314239082895076, 7.14999033169395120084430458950, 7.68410432593323419278652591373, 8.857044359692314677714524926340, 9.480525986916703007214604412159, 10.14997813198572309123652088210