L(s) = 1 | + (1.5 − 2.59i)5-s + (0.5 − 2.59i)7-s + (−2.5 − 4.33i)11-s + 2·13-s + (1 + 1.73i)17-s + (3 − 5.19i)19-s + (−1 + 1.73i)23-s + (−2 − 3.46i)25-s − 29-s + (0.5 + 0.866i)31-s + (−6 − 5.19i)35-s + (−5 + 8.66i)37-s − 4·41-s + 4·43-s + (4 − 6.92i)47-s + ⋯ |
L(s) = 1 | + (0.670 − 1.16i)5-s + (0.188 − 0.981i)7-s + (−0.753 − 1.30i)11-s + 0.554·13-s + (0.242 + 0.420i)17-s + (0.688 − 1.19i)19-s + (−0.208 + 0.361i)23-s + (−0.400 − 0.692i)25-s − 0.185·29-s + (0.0898 + 0.155i)31-s + (−1.01 − 0.878i)35-s + (−0.821 + 1.42i)37-s − 0.624·41-s + 0.609·43-s + (0.583 − 1.01i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.798490892\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.798490892\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.5 + 2.59i)T \) |
good | 5 | \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (2.5 + 4.33i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3 + 5.19i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1 - 1.73i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (5 - 8.66i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + (-4 + 6.92i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.5 - 4.33i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (6.5 + 11.2i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4 + 6.92i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-7 - 12.1i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + (-3 - 5.19i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.5 - 9.52i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 7T + 83T^{2} \) |
| 89 | \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 19T + 97T^{2} \) |
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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
−8.685204516555070348500925097758, −8.346747057119739796582791129875, −7.38534481682280194667432039612, −6.43638663487877139478220909891, −5.46896932051028645702667004199, −5.03700057765526876492813317456, −3.93798226426328505923379919355, −2.99938453640640460508747082895, −1.51261375670747518773645512088, −0.64836441377961573383700110765,
1.80777817784617706108728586105, 2.50033588674012115380288386634, 3.40943219604235664577552860876, 4.66948933101685461394439327788, 5.67423549159683224908794303725, 6.07935480851325592963968882750, 7.20361048881376163126012316129, 7.65793701233622131700668379039, 8.736691780547568924835719545975, 9.525774627127339485037275891585