L(s) = 1 | + 1.90i·2-s + i·3-s − 1.62·4-s + (−0.311 − 2.21i)5-s − 1.90·6-s + 0.719i·8-s − 9-s + (4.21 − 0.592i)10-s + 2·11-s − 1.62i·12-s + 6.42i·13-s + (2.21 − 0.311i)15-s − 4.61·16-s + 4.42i·17-s − 1.90i·18-s − 2.42·19-s + ⋯ |
L(s) = 1 | + 1.34i·2-s + 0.577i·3-s − 0.811·4-s + (−0.139 − 0.990i)5-s − 0.776·6-s + 0.254i·8-s − 0.333·9-s + (1.33 − 0.187i)10-s + 0.603·11-s − 0.468i·12-s + 1.78i·13-s + (0.571 − 0.0803i)15-s − 1.15·16-s + 1.07i·17-s − 0.448i·18-s − 0.557·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.139i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0857336 - 1.22641i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0857336 - 1.22641i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 5 | \( 1 + (0.311 + 2.21i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 1.90iT - 2T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 6.42iT - 13T^{2} \) |
| 17 | \( 1 - 4.42iT - 17T^{2} \) |
| 19 | \( 1 + 2.42T + 19T^{2} \) |
| 23 | \( 1 - 1.37iT - 23T^{2} \) |
| 29 | \( 1 + 0.755T + 29T^{2} \) |
| 31 | \( 1 + 5.18T + 31T^{2} \) |
| 37 | \( 1 + 7.61iT - 37T^{2} \) |
| 41 | \( 1 - 8.23T + 41T^{2} \) |
| 43 | \( 1 - 10.1iT - 43T^{2} \) |
| 47 | \( 1 - 2.75iT - 47T^{2} \) |
| 53 | \( 1 - 9.18iT - 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 + 6.85T + 61T^{2} \) |
| 67 | \( 1 + 2.75iT - 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 - 1.57iT - 73T^{2} \) |
| 79 | \( 1 - 4.85T + 79T^{2} \) |
| 83 | \( 1 + 11.6iT - 83T^{2} \) |
| 89 | \( 1 - 4.62T + 89T^{2} \) |
| 97 | \( 1 + 11.9iT - 97T^{2} \) |
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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
−10.87992880697985188169226920061, −9.342642690807313162078080782909, −9.122101043121774676322138266014, −8.255555820969375763094854391087, −7.34665925427669369105334865590, −6.32654004481742673525509851657, −5.66718469964334565201056478374, −4.49077159180770507744289603237, −4.04515006699795042465510280448, −1.86718664373272650919099690877,
0.62562193882476938977424344886, 2.19643238624277705314844881968, 3.02123399643056820234641464575, 3.85275289790133415522046864631, 5.37098360459983443326350085548, 6.55701797311922521624238793483, 7.26971181215233741514312860580, 8.268746775796355345442153788664, 9.399060042606303329317380467346, 10.25980881559575675385165251927