L(s) = 1 | − 1.27·2-s + 0.638i·3-s − 0.363·4-s + 1.64i·5-s − 0.816i·6-s + 4.36i·7-s + 3.02·8-s + 2.59·9-s − 2.10i·10-s + 4.08i·11-s − 0.231i·12-s − 7.00·13-s − 5.58i·14-s − 1.04·15-s − 3.14·16-s + (3.80 + 1.59i)17-s + ⋯ |
L(s) = 1 | − 0.904·2-s + 0.368i·3-s − 0.181·4-s + 0.735i·5-s − 0.333i·6-s + 1.64i·7-s + 1.06·8-s + 0.864·9-s − 0.665i·10-s + 1.23i·11-s − 0.0669i·12-s − 1.94·13-s − 1.49i·14-s − 0.270·15-s − 0.785·16-s + (0.922 + 0.387i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.922 - 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.922 - 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.144476 + 0.717530i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.144476 + 0.717530i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + (-3.80 - 1.59i)T \) |
| 43 | \( 1 + T \) |
good | 2 | \( 1 + 1.27T + 2T^{2} \) |
| 3 | \( 1 - 0.638iT - 3T^{2} \) |
| 5 | \( 1 - 1.64iT - 5T^{2} \) |
| 7 | \( 1 - 4.36iT - 7T^{2} \) |
| 11 | \( 1 - 4.08iT - 11T^{2} \) |
| 13 | \( 1 + 7.00T + 13T^{2} \) |
| 19 | \( 1 - 5.02T + 19T^{2} \) |
| 23 | \( 1 + 5.58iT - 23T^{2} \) |
| 29 | \( 1 + 3.44iT - 29T^{2} \) |
| 31 | \( 1 - 1.85iT - 31T^{2} \) |
| 37 | \( 1 - 2.73iT - 37T^{2} \) |
| 41 | \( 1 - 6.36iT - 41T^{2} \) |
| 47 | \( 1 + 8.92T + 47T^{2} \) |
| 53 | \( 1 + 6.95T + 53T^{2} \) |
| 59 | \( 1 - 3.93T + 59T^{2} \) |
| 61 | \( 1 + 11.5iT - 61T^{2} \) |
| 67 | \( 1 - 1.27T + 67T^{2} \) |
| 71 | \( 1 - 10.6iT - 71T^{2} \) |
| 73 | \( 1 + 14.0iT - 73T^{2} \) |
| 79 | \( 1 - 9.62iT - 79T^{2} \) |
| 83 | \( 1 + 4.62T + 83T^{2} \) |
| 89 | \( 1 + 5.75T + 89T^{2} \) |
| 97 | \( 1 + 7.14iT - 97T^{2} \) |
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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
−10.21553082417106536752636209318, −9.801339195314677666098385227509, −9.429519332278487034605650368319, −8.219176238621503510323732085867, −7.46821643063529977762368401166, −6.66751347553645180699416006819, −5.13104398960788668751854187956, −4.65415902396677839306213939368, −2.94696913563924314548256950212, −1.87305807047440692008029228779,
0.58381345270227968933196802667, 1.36190923667307148444799186087, 3.44962571518505930690961407322, 4.58550522690296751617182774634, 5.32946993421362001161397317189, 7.21778931715422991091167550434, 7.35086029022669099338842476976, 8.171307328916435613110627213483, 9.381376406865539547093999500031, 9.844110414382721916573142820927