L(s) = 1 | − i·2-s + (−0.711 + 1.57i)3-s − 4-s + 0.334·5-s + (1.57 + 0.711i)6-s + (−0.168 + 2.64i)7-s + i·8-s + (−1.98 − 2.24i)9-s − 0.334i·10-s − 0.616i·11-s + (0.711 − 1.57i)12-s + 3.23i·13-s + (2.64 + 0.168i)14-s + (−0.237 + 0.527i)15-s + 16-s − 3.15·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (−0.410 + 0.911i)3-s − 0.5·4-s + 0.149·5-s + (0.644 + 0.290i)6-s + (−0.0637 + 0.997i)7-s + 0.353i·8-s + (−0.662 − 0.748i)9-s − 0.105i·10-s − 0.185i·11-s + (0.205 − 0.455i)12-s + 0.898i·13-s + (0.705 + 0.0450i)14-s + (−0.0613 + 0.136i)15-s + 0.250·16-s − 0.766·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.883 - 0.467i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.883 - 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.103757 + 0.417700i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.103757 + 0.417700i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (0.711 - 1.57i)T \) |
| 7 | \( 1 + (0.168 - 2.64i)T \) |
| 23 | \( 1 + iT \) |
good | 5 | \( 1 - 0.334T + 5T^{2} \) |
| 11 | \( 1 + 0.616iT - 11T^{2} \) |
| 13 | \( 1 - 3.23iT - 13T^{2} \) |
| 17 | \( 1 + 3.15T + 17T^{2} \) |
| 19 | \( 1 + 0.890iT - 19T^{2} \) |
| 29 | \( 1 - 2.14iT - 29T^{2} \) |
| 31 | \( 1 + 2.97iT - 31T^{2} \) |
| 37 | \( 1 + 2.08T + 37T^{2} \) |
| 41 | \( 1 + 5.08T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 - 2.96T + 47T^{2} \) |
| 53 | \( 1 - 0.00920iT - 53T^{2} \) |
| 59 | \( 1 + 13.1T + 59T^{2} \) |
| 61 | \( 1 - 8.92iT - 61T^{2} \) |
| 67 | \( 1 - 5.51T + 67T^{2} \) |
| 71 | \( 1 + 4.57iT - 71T^{2} \) |
| 73 | \( 1 - 6.21iT - 73T^{2} \) |
| 79 | \( 1 + 7.38T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 + 15.6T + 89T^{2} \) |
| 97 | \( 1 - 10.2iT - 97T^{2} \) |
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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
−10.36134019986556302653932095260, −9.571038090493476683324977373574, −9.002561582722671730895532602443, −8.317407931030418799067292148118, −6.73417967728748973949470104186, −5.86779342694488667285213498804, −4.99907078643048931286286923062, −4.15211135041459082772540965073, −3.08918928373789181853519038237, −1.94861262722800559384426337396,
0.20363313381305669482445988482, 1.69277834093558783755798011693, 3.31086985040550957072321971989, 4.59233025082258813861368761530, 5.50884262813503044455401678469, 6.41532082901820211797380609623, 7.06445038258317238079437296512, 7.84391202661622327582330259919, 8.454531261984056700027417290878, 9.701078560529822369027299789092