L(s) = 1 | + 4.22·5-s + (−2.37 + 1.15i)7-s − 1.92·11-s + (−0.291 + 0.504i)13-s + (−3.61 + 6.25i)17-s + (2.10 + 3.64i)19-s − 1.27·23-s + 12.8·25-s + (4.20 + 7.27i)29-s + (0.476 + 0.824i)31-s + (−10.0 + 4.89i)35-s + (3.03 + 5.25i)37-s + (−1.31 + 2.27i)41-s + (0.442 + 0.766i)43-s + (2.88 − 4.99i)47-s + ⋯ |
L(s) = 1 | + 1.88·5-s + (−0.898 + 0.438i)7-s − 0.581·11-s + (−0.0808 + 0.140i)13-s + (−0.875 + 1.51i)17-s + (0.482 + 0.835i)19-s − 0.266·23-s + 2.56·25-s + (0.780 + 1.35i)29-s + (0.0855 + 0.148i)31-s + (−1.69 + 0.827i)35-s + (0.498 + 0.863i)37-s + (−0.205 + 0.355i)41-s + (0.0674 + 0.116i)43-s + (0.420 − 0.728i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.394 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.394 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.915570599\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.915570599\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.37 - 1.15i)T \) |
good | 5 | \( 1 - 4.22T + 5T^{2} \) |
| 11 | \( 1 + 1.92T + 11T^{2} \) |
| 13 | \( 1 + (0.291 - 0.504i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (3.61 - 6.25i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.10 - 3.64i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 1.27T + 23T^{2} \) |
| 29 | \( 1 + (-4.20 - 7.27i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.476 - 0.824i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.03 - 5.25i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.31 - 2.27i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.442 - 0.766i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.88 + 4.99i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.962 + 1.66i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2.27 + 3.94i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.29 + 9.16i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.43 - 4.21i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 11.5T + 71T^{2} \) |
| 73 | \( 1 + (-0.446 + 0.772i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.93 + 10.2i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.24 - 9.08i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (3.87 + 6.71i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.98 + 3.44i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.741917689320303755287151599101, −8.925451225936943455457314891995, −8.280726647898690007136989232815, −6.86351294219996424111023803327, −6.26801403149444941133537572550, −5.69789811916221710758920414900, −4.84700586411013090511000933098, −3.40364142763545849704262184249, −2.43036988329813666206768337318, −1.56091094939532692415349781025,
0.73781256579611049972225157487, 2.40310488380015957432453961569, 2.80253872111845770912302258195, 4.41480217896154558519691439956, 5.32171977631942622652793844781, 6.06218853181166047471611783529, 6.78141516727363160739537614897, 7.51655143129826484752571946972, 8.907759106276987893352161196112, 9.412220947794757473079086467761