L(s) = 1 | + (2 − 2.23i)3-s − 8·7-s + (−1.00 − 8.94i)9-s + 8.94i·11-s + 12·13-s + 31.3i·17-s − 6·19-s + (−16 + 17.8i)21-s − 4.47i·23-s + (−22.0 − 15.6i)27-s + 26.8i·29-s − 34·31-s + (20.0 + 17.8i)33-s + 44·37-s + (24 − 26.8i)39-s + ⋯ |
L(s) = 1 | + (0.666 − 0.745i)3-s − 1.14·7-s + (−0.111 − 0.993i)9-s + 0.813i·11-s + 0.923·13-s + 1.84i·17-s − 0.315·19-s + (−0.761 + 0.851i)21-s − 0.194i·23-s + (−0.814 − 0.579i)27-s + 0.925i·29-s − 1.09·31-s + (0.606 + 0.542i)33-s + 1.18·37-s + (0.615 − 0.688i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.666 - 0.745i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.666 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.672305201\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.672305201\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2 + 2.23i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 8T + 49T^{2} \) |
| 11 | \( 1 - 8.94iT - 121T^{2} \) |
| 13 | \( 1 - 12T + 169T^{2} \) |
| 17 | \( 1 - 31.3iT - 289T^{2} \) |
| 19 | \( 1 + 6T + 361T^{2} \) |
| 23 | \( 1 + 4.47iT - 529T^{2} \) |
| 29 | \( 1 - 26.8iT - 841T^{2} \) |
| 31 | \( 1 + 34T + 961T^{2} \) |
| 37 | \( 1 - 44T + 1.36e3T^{2} \) |
| 41 | \( 1 - 17.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 28T + 1.84e3T^{2} \) |
| 47 | \( 1 - 4.47iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 40.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 98.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 74T + 3.72e3T^{2} \) |
| 67 | \( 1 - 92T + 4.48e3T^{2} \) |
| 71 | \( 1 + 53.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 56T + 5.32e3T^{2} \) |
| 79 | \( 1 + 78T + 6.24e3T^{2} \) |
| 83 | \( 1 - 102. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 17.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 32T + 9.40e3T^{2} \) |
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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
−9.517256147066829449582634919801, −8.810979434028453830621339879756, −8.080436648943174777708516971532, −7.14525525634516896475142155311, −6.41505152691108645914084264257, −5.80737177424087145750418558197, −4.12952749495053036363055055534, −3.46950251720632822980462670553, −2.33125157415396602263468718351, −1.23302541003468110864519699742,
0.47680570151828622420258285760, 2.40537621455211943689836759981, 3.29591769873658924393318082142, 3.95831363824835224703924530203, 5.16594075755785682266867686534, 6.03198557976057291601956420863, 6.99233446697566678145426715346, 7.975294939684548847343027349580, 8.808050031617336576180631260548, 9.491362590206890531935289695882