| L(s) = 1 | − 5.73i·5-s + 2.64i·7-s − 1.40·11-s − 19.0i·13-s + 32.2·17-s − 12.5·19-s + 15.8i·23-s − 7.86·25-s + 3.29i·29-s − 22.6i·31-s + 15.1·35-s − 54.1i·37-s + 7.59·41-s + 20.8·43-s + 21.6i·47-s + ⋯ |
| L(s) = 1 | − 1.14i·5-s + 0.377i·7-s − 0.127·11-s − 1.46i·13-s + 1.89·17-s − 0.661·19-s + 0.690i·23-s − 0.314·25-s + 0.113i·29-s − 0.731i·31-s + 0.433·35-s − 1.46i·37-s + 0.185·41-s + 0.484·43-s + 0.460i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.546 + 0.837i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.546 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.659717867\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.659717867\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
| good | 5 | \( 1 + 5.73iT - 25T^{2} \) |
| 11 | \( 1 + 1.40T + 121T^{2} \) |
| 13 | \( 1 + 19.0iT - 169T^{2} \) |
| 17 | \( 1 - 32.2T + 289T^{2} \) |
| 19 | \( 1 + 12.5T + 361T^{2} \) |
| 23 | \( 1 - 15.8iT - 529T^{2} \) |
| 29 | \( 1 - 3.29iT - 841T^{2} \) |
| 31 | \( 1 + 22.6iT - 961T^{2} \) |
| 37 | \( 1 + 54.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 7.59T + 1.68e3T^{2} \) |
| 43 | \( 1 - 20.8T + 1.84e3T^{2} \) |
| 47 | \( 1 - 21.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 0.356iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 26.8T + 3.48e3T^{2} \) |
| 61 | \( 1 - 86.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 114.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 104. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 24.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + 117. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 79.2T + 6.88e3T^{2} \) |
| 89 | \( 1 + 2.66T + 7.92e3T^{2} \) |
| 97 | \( 1 + 52.0T + 9.40e3T^{2} \) |
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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
−8.731417193480547669188830573687, −7.86158664291930360118682051410, −7.50816427785067567625195073361, −5.90208857465768145567695154098, −5.61580350303143028118096466752, −4.77425677801959615192870279319, −3.71077607777395106720387781446, −2.76411357664872317780444643687, −1.41800380540568796716613007174, −0.45180366703321534338482215485,
1.27896221416979120014107971004, 2.48923456702483553819259175535, 3.37600615895400312589535923754, 4.22255026197519502527021259351, 5.23150998888419314418303905418, 6.36649884963145450504681607049, 6.77883924411849736536591099316, 7.58397201741010793745670723267, 8.368309724121058978119181084159, 9.328217425946668292332482725811
