| L(s) = 1 | + 4.58i·5-s + (3.81 − 5.87i)7-s + 2.74i·11-s + 23.8·13-s − 3.77·17-s − 22.4i·19-s + 6.70·23-s + 3.99·25-s − 30.2·29-s + 11.6·31-s + (26.9 + 17.4i)35-s − 1.51i·37-s + 3.87·41-s + 39.3·43-s − 86.1i·47-s + ⋯ |
| L(s) = 1 | + 0.916i·5-s + (0.544 − 0.838i)7-s + 0.249i·11-s + 1.83·13-s − 0.221·17-s − 1.18i·19-s + 0.291·23-s + 0.159·25-s − 1.04·29-s + 0.377·31-s + (0.768 + 0.499i)35-s − 0.0408i·37-s + 0.0945·41-s + 0.915·43-s − 1.83i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 + 0.302i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.953 + 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.407592420\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.407592420\) |
| \(L(2)\) |
|
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 | \( 1 + (-3.81 + 5.87i)T \) |
| good | 5 | \( 1 - 4.58iT - 25T^{2} \) |
| 11 | \( 1 - 2.74iT - 121T^{2} \) |
| 13 | \( 1 - 23.8T + 169T^{2} \) |
| 17 | \( 1 + 3.77T + 289T^{2} \) |
| 19 | \( 1 + 22.4iT - 361T^{2} \) |
| 23 | \( 1 - 6.70T + 529T^{2} \) |
| 29 | \( 1 + 30.2T + 841T^{2} \) |
| 31 | \( 1 - 11.6T + 961T^{2} \) |
| 37 | \( 1 + 1.51iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 3.87T + 1.68e3T^{2} \) |
| 43 | \( 1 - 39.3T + 1.84e3T^{2} \) |
| 47 | \( 1 + 86.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 56.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + 37.8T + 3.48e3T^{2} \) |
| 61 | \( 1 + 66.4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 79.4T + 4.48e3T^{2} \) |
| 71 | \( 1 - 98.3T + 5.04e3T^{2} \) |
| 73 | \( 1 - 114. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 42.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 62.5T + 6.88e3T^{2} \) |
| 89 | \( 1 - 164.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 97.0iT - 9.40e3T^{2} \) |
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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
−8.855478758106483871724473816353, −8.145530122969940590402749593152, −7.19813773308813501348274886256, −6.74766400752004352396816943138, −5.87045581517494055470080806418, −4.79008353821096044096295467774, −3.89864050303101418052562693571, −3.14658409457805301673777704124, −1.92069597068720910340533338843, −0.74697565036628383026578133354,
1.03183812820416351680471976185, 1.82022684758849517191098425844, 3.19396952980882723126650986376, 4.14004006745723266001341622774, 5.02655785758561770679392666233, 5.86686678956957740416472693624, 6.33401237671406319585976562765, 7.82876754274114030033189677244, 8.215966355665689557936246591700, 9.048150031251927250010959434151
