| L(s) = 1 | + (2 − 2.23i)3-s − 2.23i·5-s − 6·7-s + (−1.00 − 8.94i)9-s − 4.47i·11-s − 16·13-s + (−5.00 − 4.47i)15-s + 4.47i·17-s + 2·19-s + (−12 + 13.4i)21-s + 13.4i·23-s − 5.00·25-s + (−22.0 − 15.6i)27-s + 31.3i·29-s − 18·31-s + ⋯ |
| L(s) = 1 | + (0.666 − 0.745i)3-s − 0.447i·5-s − 0.857·7-s + (−0.111 − 0.993i)9-s − 0.406i·11-s − 1.23·13-s + (−0.333 − 0.298i)15-s + 0.263i·17-s + 0.105·19-s + (−0.571 + 0.638i)21-s + 0.583i·23-s − 0.200·25-s + (−0.814 − 0.579i)27-s + 1.07i·29-s − 0.580·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.745 - 0.666i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.745 - 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3173660864\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3173660864\) |
| \(L(2)\) |
|
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 | \( 1 + 2.23iT \) |
| 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
−9.369317987145357652346132520001, −8.453390269387523133897739370828, −7.64047378899354147729507971155, −6.87044966700301108406721892460, −6.02148426302105952664043032474, −4.93326657896860365665155192059, −3.60653301888819599684046253340, −2.79605874642251400436888695018, −1.53133951491367683391982936359, −0.086027512788810667266821994792,
2.25058790495281515450366470130, 3.00132364844718114503731334732, 4.05494594856040619999608653792, 4.93971733998390532127082058106, 6.05577318849746578906627147384, 7.15067921766866296835217335077, 7.75737669104397396143380662810, 8.909903037297951816079553736473, 9.632789661362157763880760345370, 10.10072915033844897428547874623
