L(s) = 1 | + 1.41·3-s + (1.41 + 1.73i)5-s − 2.44i·7-s − 0.999·9-s + 3.46i·11-s + (2.00 + 2.44i)15-s − 4.89i·17-s − 3.46i·19-s − 3.46i·21-s + 2.44i·23-s + (−0.999 + 4.89i)25-s − 5.65·27-s − 4·31-s + 4.89i·33-s + (4.24 − 3.46i)35-s + ⋯ |
L(s) = 1 | + 0.816·3-s + (0.632 + 0.774i)5-s − 0.925i·7-s − 0.333·9-s + 1.04i·11-s + (0.516 + 0.632i)15-s − 1.18i·17-s − 0.794i·19-s − 0.755i·21-s + 0.510i·23-s + (−0.199 + 0.979i)25-s − 1.08·27-s − 0.718·31-s + 0.852i·33-s + (0.717 − 0.585i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.160i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.46786 + 0.118507i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.46786 + 0.118507i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.41 - 1.73i)T \) |
good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 7 | \( 1 + 2.44iT - 7T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 4.89iT - 17T^{2} \) |
| 19 | \( 1 + 3.46iT - 19T^{2} \) |
| 23 | \( 1 - 2.44iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 8.48T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 4.24T + 43T^{2} \) |
| 47 | \( 1 + 7.34iT - 47T^{2} \) |
| 53 | \( 1 - 5.65T + 53T^{2} \) |
| 59 | \( 1 - 10.3iT - 59T^{2} \) |
| 61 | \( 1 - 3.46iT - 61T^{2} \) |
| 67 | \( 1 - 4.24T + 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 4.89iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 9.89T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 4.89iT - 97T^{2} \) |
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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
−13.40578010665427744095130949998, −11.85802085269407896740951270716, −10.75281146162890974282977055369, −9.822767102708738431453399301015, −8.987713305828881390722723649643, −7.48179346508146734748482155634, −6.88400947351338394806531387559, −5.20126341299700026547296509883, −3.56802123144697703378578358297, −2.27577330372620020553451553834,
2.05406325523293706674455829453, 3.54690223371836310274677125087, 5.37306533955832967098114330914, 6.18496847811091275157070007980, 8.224443468460717456966353242797, 8.610490931788288429176969281261, 9.530617812293066640554114578445, 10.80214363082595556972974455849, 12.10126541236893761073302719462, 12.91263128160951214659368008086