| L(s) = 1 | − 1.41·2-s + 2.00·4-s + 6.63i·5-s − 2.82·8-s − 9.37i·10-s + 4.75·11-s + 15.2i·13-s + 4.00·16-s − 3.76i·17-s + 4.18i·19-s + 13.2i·20-s − 6.72·22-s + 27.7·23-s − 18.9·25-s − 21.6i·26-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.500·4-s + 1.32i·5-s − 0.353·8-s − 0.937i·10-s + 0.432·11-s + 1.17i·13-s + 0.250·16-s − 0.221i·17-s + 0.220i·19-s + 0.663i·20-s − 0.305·22-s + 1.20·23-s − 0.758·25-s − 0.831i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.034730390\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.034730390\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 6.63iT - 25T^{2} \) |
| 11 | \( 1 - 4.75T + 121T^{2} \) |
| 13 | \( 1 - 15.2iT - 169T^{2} \) |
| 17 | \( 1 + 3.76iT - 289T^{2} \) |
| 19 | \( 1 - 4.18iT - 361T^{2} \) |
| 23 | \( 1 - 27.7T + 529T^{2} \) |
| 29 | \( 1 + 3.51T + 841T^{2} \) |
| 31 | \( 1 - 48.8iT - 961T^{2} \) |
| 37 | \( 1 + 2.94T + 1.36e3T^{2} \) |
| 41 | \( 1 - 27.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 10.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + 52.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 55.9T + 2.80e3T^{2} \) |
| 59 | \( 1 - 38.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 90.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 34.6T + 4.48e3T^{2} \) |
| 71 | \( 1 + 36.4T + 5.04e3T^{2} \) |
| 73 | \( 1 + 52.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 33.7T + 6.24e3T^{2} \) |
| 83 | \( 1 - 127. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 50.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 101. iT - 9.40e3T^{2} \) |
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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
−10.29369789568810792141580521755, −9.411810161259599016055813476086, −8.736507289472408168481807934508, −7.61506599739505022399761333989, −6.76726558776477758564886374600, −6.48380343878567792125329679346, −5.02431952444926802774227512701, −3.64837622238910515353997446806, −2.73148707238963296827553810428, −1.50288773400785590953261439620,
0.45597250173844163118861628923, 1.44320860688683736308286188994, 2.92739685115207037772064566134, 4.26808845940740115553345564710, 5.27504741678829683268558853354, 6.09732967932960334635860894503, 7.32126247309450382623160127347, 8.086753391177903346814184793535, 8.856404166612702548367102307083, 9.403847132237843110476284700040
