| L(s) = 1 | − 1.41i·2-s − 1.73·3-s − 2.00·4-s + 2.44i·6-s + (−3.49 − 6.06i)7-s + 2.82i·8-s + 2.99·9-s − 17.5·11-s + 3.46·12-s + 17.6·13-s + (−8.57 + 4.94i)14-s + 4.00·16-s − 14.0·17-s − 4.24i·18-s + 16.6i·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577·3-s − 0.500·4-s + 0.408i·6-s + (−0.499 − 0.866i)7-s + 0.353i·8-s + 0.333·9-s − 1.59·11-s + 0.288·12-s + 1.35·13-s + (−0.612 + 0.353i)14-s + 0.250·16-s − 0.828·17-s − 0.235i·18-s + 0.873i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0598i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7927253497\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7927253497\) |
| \(L(2)\) |
|
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 | \( 1 + 1.73T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (3.49 + 6.06i)T \) |
| good | 11 | \( 1 + 17.5T + 121T^{2} \) |
| 13 | \( 1 - 17.6T + 169T^{2} \) |
| 17 | \( 1 + 14.0T + 289T^{2} \) |
| 19 | \( 1 - 16.6iT - 361T^{2} \) |
| 23 | \( 1 - 15.5iT - 529T^{2} \) |
| 29 | \( 1 + 36.9T + 841T^{2} \) |
| 31 | \( 1 + 54.2iT - 961T^{2} \) |
| 37 | \( 1 - 45.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 1.20iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 50.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 91.7T + 2.20e3T^{2} \) |
| 53 | \( 1 - 51.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 14.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 91.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 5.61iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 80.8T + 5.04e3T^{2} \) |
| 73 | \( 1 - 91.0T + 5.32e3T^{2} \) |
| 79 | \( 1 - 28.5T + 6.24e3T^{2} \) |
| 83 | \( 1 - 50.3T + 6.88e3T^{2} \) |
| 89 | \( 1 - 76.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 125.T + 9.40e3T^{2} \) |
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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
−10.00912446084297482582654939700, −9.111928217194540670429533306237, −8.025605581165315178285952210574, −7.33744951882169925968397422079, −6.11656592132162365737326034960, −5.46309916055324189755847787099, −4.23054645256129042559453880622, −3.55394928848041695126880898104, −2.21230194307257145679305764870, −0.820644442807828280987326355252,
0.36943779726872188045027399908, 2.27786919667193953379056915450, 3.52367006939367429042732129597, 4.83943471929632398329602814483, 5.50151795913379842237916960669, 6.27888900470298325738399277781, 7.01508136590743961052386146226, 8.079859318988136747589280505073, 8.811919535166761448665429630315, 9.507979492638496229792703933858
