| L(s) = 1 | − 1.41·2-s − 1.73i·3-s + 2.00·4-s + 2.44i·6-s + (5.11 + 4.78i)7-s − 2.82·8-s − 2.99·9-s + 10.3·11-s − 3.46i·12-s + 0.605i·13-s + (−7.23 − 6.76i)14-s + 4.00·16-s − 5.49i·17-s + 4.24·18-s + 33.2i·19-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577i·3-s + 0.500·4-s + 0.408i·6-s + (0.730 + 0.683i)7-s − 0.353·8-s − 0.333·9-s + 0.941·11-s − 0.288i·12-s + 0.0465i·13-s + (−0.516 − 0.483i)14-s + 0.250·16-s − 0.323i·17-s + 0.235·18-s + 1.74i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.683 - 0.730i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.683 - 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.364328504\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.364328504\) |
| \(L(2)\) |
|
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 | \( 1 + 1.73iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-5.11 - 4.78i)T \) |
| good | 11 | \( 1 - 10.3T + 121T^{2} \) |
| 13 | \( 1 - 0.605iT - 169T^{2} \) |
| 17 | \( 1 + 5.49iT - 289T^{2} \) |
| 19 | \( 1 - 33.2iT - 361T^{2} \) |
| 23 | \( 1 + 14.4T + 529T^{2} \) |
| 29 | \( 1 - 42.0T + 841T^{2} \) |
| 31 | \( 1 + 0.540iT - 961T^{2} \) |
| 37 | \( 1 + 13.6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 13.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 82.3T + 1.84e3T^{2} \) |
| 47 | \( 1 - 53.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 19.4T + 2.80e3T^{2} \) |
| 59 | \( 1 - 29.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 74.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 12.1T + 4.48e3T^{2} \) |
| 71 | \( 1 - 42.3T + 5.04e3T^{2} \) |
| 73 | \( 1 + 66.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 27.0T + 6.24e3T^{2} \) |
| 83 | \( 1 + 126. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 30.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 164. iT - 9.40e3T^{2} \) |
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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
−9.772880651717006597878409330184, −8.804805771310277996168992519115, −8.249304047868001548696881718958, −7.53374114888448879533254819846, −6.46323963976624702260648982626, −5.87281953859475755810605658030, −4.66133596846589402825409120468, −3.31700120505257091261383009772, −2.02779829212805491970175317396, −1.22053153724697357343000299446,
0.59016090939409465109001854722, 1.89051779951353420598727109308, 3.29102850332448453402686688704, 4.34604357075022191791715760635, 5.15728516749731243191694516546, 6.51032867016588770680819527090, 7.07156238784695967868513095509, 8.268638022947232560628599815759, 8.690964402982236104205787923554, 9.715939173112572047871703937698
