| L(s) = 1 | + (−2 + 2.23i)3-s − 6·7-s + (−1.00 − 8.94i)9-s − 4.47i·11-s − 16·13-s − 4.47i·17-s + 2·19-s + (12 − 13.4i)21-s + 13.4i·23-s + (22.0 + 15.6i)27-s − 31.3i·29-s + 18·31-s + (10.0 + 8.94i)33-s + 16·37-s + (32 − 35.7i)39-s + ⋯ |
| L(s) = 1 | + (−0.666 + 0.745i)3-s − 0.857·7-s + (−0.111 − 0.993i)9-s − 0.406i·11-s − 1.23·13-s − 0.263i·17-s + 0.105·19-s + (0.571 − 0.638i)21-s + 0.583i·23-s + (0.814 + 0.579i)27-s − 1.07i·29-s + 0.580·31-s + (0.303 + 0.271i)33-s + 0.432·37-s + (0.820 − 0.917i)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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9308474905\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9308474905\) |
| \(L(2)\) |
|
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 | \( 1 + 6T + 49T^{2} \) |
| 11 | \( 1 + 4.47iT - 121T^{2} \) |
| 13 | \( 1 + 16T + 169T^{2} \) |
| 17 | \( 1 + 4.47iT - 289T^{2} \) |
| 19 | \( 1 - 2T + 361T^{2} \) |
| 23 | \( 1 - 13.4iT - 529T^{2} \) |
| 29 | \( 1 + 31.3iT - 841T^{2} \) |
| 31 | \( 1 - 18T + 961T^{2} \) |
| 37 | \( 1 - 16T + 1.36e3T^{2} \) |
| 41 | \( 1 - 62.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 16T + 1.84e3T^{2} \) |
| 47 | \( 1 + 49.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 4.47iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 4.47iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 82T + 3.72e3T^{2} \) |
| 67 | \( 1 - 24T + 4.48e3T^{2} \) |
| 71 | \( 1 - 125. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 74T + 5.32e3T^{2} \) |
| 79 | \( 1 + 138T + 6.24e3T^{2} \) |
| 83 | \( 1 + 93.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 107. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 166T + 9.40e3T^{2} \) |
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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
−9.870189956105534740342079379953, −9.152138843053301613219544213786, −8.053067286814975400494582038158, −7.00364697769062200555444437153, −6.24722051673095965967434109595, −5.41555531322977223089938423849, −4.55958023263356091398041598008, −3.57046322076794530510793402974, −2.59732113995133207114308369870, −0.64831781437864250286717826362,
0.51045805984927945135688069233, 1.99361371667670981181193745280, 2.99725238673989170812682321085, 4.42400271527483857310422251350, 5.29433910740222920370483027020, 6.21454994697034955952822998534, 6.96783834518912322461383661543, 7.53852238838885336684115661584, 8.569186986110591650901523041040, 9.588179021561948133725661506455
