L(s) = 1 | + (1 − 1.73i)5-s + (−2.5 + 0.866i)7-s + (3 + 5.19i)11-s − 3·13-s + (2 + 3.46i)17-s + (−2.5 + 4.33i)19-s + (2 − 3.46i)23-s + (0.500 + 0.866i)25-s + 4·29-s + (3.5 + 6.06i)31-s + (−1.00 + 5.19i)35-s + (4.5 − 7.79i)37-s + 2·41-s + 43-s + (−1 + 1.73i)47-s + ⋯ |
L(s) = 1 | + (0.447 − 0.774i)5-s + (−0.944 + 0.327i)7-s + (0.904 + 1.56i)11-s − 0.832·13-s + (0.485 + 0.840i)17-s + (−0.573 + 0.993i)19-s + (0.417 − 0.722i)23-s + (0.100 + 0.173i)25-s + 0.742·29-s + (0.628 + 1.08i)31-s + (−0.169 + 0.878i)35-s + (0.739 − 1.28i)37-s + 0.312·41-s + 0.152·43-s + (−0.145 + 0.252i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.443034865\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.443034865\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.5 - 0.866i)T \) |
good | 5 | \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 3T + 13T^{2} \) |
| 17 | \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 + (-3.5 - 6.06i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.5 + 7.79i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 + (1 - 1.73i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4 - 6.92i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (7.5 + 12.9i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + (-5.5 - 9.52i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + (4 - 6.92i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−9.947016432548702552357678611376, −9.322889715259796338036299168142, −8.657473803990557409709855779032, −7.50765665145583404753682417190, −6.63492407878902111299673062076, −5.86642245507459681783434218931, −4.79284101767179958778806531719, −3.99649392325846079452411594199, −2.57874523151774041534991501304, −1.41258495170933294239625370695,
0.70667779511792272747013891799, 2.68472768168333848800309426797, 3.24351860671157406808041880531, 4.49683100950254419439239739680, 5.77973276086698090919173261824, 6.51333271219226954514676867315, 7.05789296144251081477554203990, 8.205687515322217319418128667470, 9.247718076365400481914575016681, 9.769858902041848524187846930721