| L(s) = 1 | − 1.41i·2-s + (2.79 − 1.07i)3-s − 2.00·4-s + (−1.52 − 3.95i)6-s − 2.64·7-s + 2.82i·8-s + (6.67 − 6.03i)9-s + 5.98i·11-s + (−5.59 + 2.15i)12-s + 19.0·13-s + 3.74i·14-s + 4.00·16-s + 2.02i·17-s + (−8.53 − 9.44i)18-s + 21.1·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + (0.933 − 0.359i)3-s − 0.500·4-s + (−0.253 − 0.659i)6-s − 0.377·7-s + 0.353i·8-s + (0.742 − 0.670i)9-s + 0.544i·11-s + (−0.466 + 0.179i)12-s + 1.46·13-s + 0.267i·14-s + 0.250·16-s + 0.119i·17-s + (−0.473 − 0.524i)18-s + 1.11·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.359 + 0.933i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.359 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.775177553\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.775177553\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 + (-2.79 + 1.07i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 2.64T \) |
| good | 11 | \( 1 - 5.98iT - 121T^{2} \) |
| 13 | \( 1 - 19.0T + 169T^{2} \) |
| 17 | \( 1 - 2.02iT - 289T^{2} \) |
| 19 | \( 1 - 21.1T + 361T^{2} \) |
| 23 | \( 1 - 32.2iT - 529T^{2} \) |
| 29 | \( 1 + 12.2iT - 841T^{2} \) |
| 31 | \( 1 - 39.3T + 961T^{2} \) |
| 37 | \( 1 + 20.3T + 1.36e3T^{2} \) |
| 41 | \( 1 - 34.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 40.3T + 1.84e3T^{2} \) |
| 47 | \( 1 + 72.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 57.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 71.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 81.7T + 3.72e3T^{2} \) |
| 67 | \( 1 - 81.7T + 4.48e3T^{2} \) |
| 71 | \( 1 + 7.70iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 12.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + 131.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 115. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 101. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 8.19T + 9.40e3T^{2} \) |
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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
−9.800910345387673093616957981349, −8.727261428621796334417759686565, −8.152705630999938327968464933802, −7.17189730605788546393227831298, −6.26547064576556442233780582258, −5.05977496835717003338621521799, −3.74326567949301406796751615095, −3.30445688581841150016822272950, −2.01978224491117023525067010346, −1.02898188985352156451739991474,
1.08384221455879069700641657957, 2.81522284205856705585058620531, 3.64597344918205374358646139495, 4.57708197223845993225967634711, 5.68946234002692327558437801127, 6.57228083434377962996896546531, 7.44228733247310698091825744030, 8.528879007849227888982932312726, 8.660565826388692938737596236902, 9.722450537204938969676829918354
