L(s) = 1 | − i·3-s + (1.48 − 1.67i)5-s + i·7-s − 9-s + 6.31·11-s + 6.96i·13-s + (−1.67 − 1.48i)15-s − 6.57i·17-s + 3.73·19-s + 21-s + 5.73i·23-s + (−0.612 − 4.96i)25-s + i·27-s + 2·29-s − 1.03·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + (0.662 − 0.749i)5-s + 0.377i·7-s − 0.333·9-s + 1.90·11-s + 1.93i·13-s + (−0.432 − 0.382i)15-s − 1.59i·17-s + 0.857·19-s + 0.218·21-s + 1.19i·23-s + (−0.122 − 0.992i)25-s + 0.192i·27-s + 0.371·29-s − 0.186·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.80414 - 0.683242i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.80414 - 0.683242i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (-1.48 + 1.67i)T \) |
| 7 | \( 1 - iT \) |
good | 11 | \( 1 - 6.31T + 11T^{2} \) |
| 13 | \( 1 - 6.96iT - 13T^{2} \) |
| 17 | \( 1 + 6.57iT - 17T^{2} \) |
| 19 | \( 1 - 3.73T + 19T^{2} \) |
| 23 | \( 1 - 5.73iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 1.03T + 31T^{2} \) |
| 37 | \( 1 + 10.7iT - 37T^{2} \) |
| 41 | \( 1 + 6.96T + 41T^{2} \) |
| 43 | \( 1 + 5.92iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 1.03iT - 53T^{2} \) |
| 59 | \( 1 - 3.22T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 - 4.77iT - 67T^{2} \) |
| 71 | \( 1 - 8.23T + 71T^{2} \) |
| 73 | \( 1 - 4.26iT - 73T^{2} \) |
| 79 | \( 1 - 5.92T + 79T^{2} \) |
| 83 | \( 1 - 3.22iT - 83T^{2} \) |
| 89 | \( 1 + 2.18T + 89T^{2} \) |
| 97 | \( 1 + 3.73iT - 97T^{2} \) |
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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
−9.578486471816036648640419327295, −9.297903261198520460382124900075, −8.751112078144461954424793281652, −7.29423806903078416225546235738, −6.72629861712198075722114391934, −5.78514580328316906270885097271, −4.81998918884928241872155186892, −3.72663849551707956447343889321, −2.11412961738966274410127478952, −1.22736369760018781660192023016,
1.36151067497188397762866344712, 3.03815599890933320957317920990, 3.73835875176629263664898565403, 4.96084861583235354830568164522, 6.13530627668238524049406606586, 6.54248915374155041465876171311, 7.83785398821685713588484082068, 8.675763376275315525465793646313, 9.694058932568530371278911765726, 10.29564324884027229897433882874