L(s) = 1 | − 2.41·3-s + (1.94 + 1.10i)5-s + i·7-s + 2.84·9-s + 2.55i·11-s + 5.47·13-s + (−4.70 − 2.66i)15-s − 2.44i·17-s + 4.99i·19-s − 2.41i·21-s − 1.10i·23-s + (2.56 + 4.29i)25-s + 0.376·27-s − 1.96i·29-s + 2.33·31-s + ⋯ |
L(s) = 1 | − 1.39·3-s + (0.869 + 0.493i)5-s + 0.377i·7-s + 0.948·9-s + 0.770i·11-s + 1.51·13-s + (−1.21 − 0.688i)15-s − 0.593i·17-s + 1.14i·19-s − 0.527i·21-s − 0.229i·23-s + (0.513 + 0.858i)25-s + 0.0724·27-s − 0.364i·29-s + 0.419·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.251 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.251 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.298754442\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.298754442\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.94 - 1.10i)T \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 + 2.41T + 3T^{2} \) |
| 11 | \( 1 - 2.55iT - 11T^{2} \) |
| 13 | \( 1 - 5.47T + 13T^{2} \) |
| 17 | \( 1 + 2.44iT - 17T^{2} \) |
| 19 | \( 1 - 4.99iT - 19T^{2} \) |
| 23 | \( 1 + 1.10iT - 23T^{2} \) |
| 29 | \( 1 + 1.96iT - 29T^{2} \) |
| 31 | \( 1 - 2.33T + 31T^{2} \) |
| 37 | \( 1 - 6.16T + 37T^{2} \) |
| 41 | \( 1 + 9.40T + 41T^{2} \) |
| 43 | \( 1 - 5.99T + 43T^{2} \) |
| 47 | \( 1 + 8.05iT - 47T^{2} \) |
| 53 | \( 1 + 1.97T + 53T^{2} \) |
| 59 | \( 1 + 1.44iT - 59T^{2} \) |
| 61 | \( 1 - 8.71iT - 61T^{2} \) |
| 67 | \( 1 + 3.92T + 67T^{2} \) |
| 71 | \( 1 - 4.55T + 71T^{2} \) |
| 73 | \( 1 + 7.55iT - 73T^{2} \) |
| 79 | \( 1 + 8.53T + 79T^{2} \) |
| 83 | \( 1 + 8.26T + 83T^{2} \) |
| 89 | \( 1 - 17.2T + 89T^{2} \) |
| 97 | \( 1 - 10.6iT - 97T^{2} \) |
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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
−9.352815514037563600951160546194, −8.494514487839309986143076503718, −7.42970046523828454206406025279, −6.50161798679697628013560526245, −6.09742438158100604574936319413, −5.46414333764542602998819189873, −4.62663095917852206908961818138, −3.48782450955038475245635809633, −2.21097398494966109943936442513, −1.13099823508839018868815643218,
0.66529181140493029095840844609, 1.49204519255626513338578114278, 3.06330298843996381011817837173, 4.26850187092300831864659666713, 5.03460230725007850391514042005, 5.95802940480424452250855756595, 6.15327471966795461335318640489, 7.01494722211383441577540484886, 8.279037050593886563207138291007, 8.845882866689111680057280530912