L(s) = 1 | + (−0.5 + 0.866i)5-s + (0.835 + 2.51i)7-s + (3.11 + 5.39i)11-s − 3.77·13-s + (0.313 + 0.542i)17-s + (−0.206 + 0.357i)19-s + (2.04 − 3.53i)23-s + (−0.499 − 0.866i)25-s − 4.12·29-s + (4.32 + 7.49i)31-s + (−2.59 − 0.532i)35-s + (−3.59 + 6.22i)37-s + 4.88·41-s − 10.6·43-s + (5.49 − 9.52i)47-s + ⋯ |
L(s) = 1 | + (−0.223 + 0.387i)5-s + (0.315 + 0.948i)7-s + (0.938 + 1.62i)11-s − 1.04·13-s + (0.0759 + 0.131i)17-s + (−0.0474 + 0.0821i)19-s + (0.425 − 0.737i)23-s + (−0.0999 − 0.173i)25-s − 0.766·29-s + (0.777 + 1.34i)31-s + (−0.438 − 0.0899i)35-s + (−0.590 + 1.02i)37-s + 0.763·41-s − 1.62·43-s + (0.802 − 1.38i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 - 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.276436645\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.276436645\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (-0.835 - 2.51i)T \) |
good | 11 | \( 1 + (-3.11 - 5.39i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 3.77T + 13T^{2} \) |
| 17 | \( 1 + (-0.313 - 0.542i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.206 - 0.357i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.04 + 3.53i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 4.12T + 29T^{2} \) |
| 31 | \( 1 + (-4.32 - 7.49i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.59 - 6.22i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 4.88T + 41T^{2} \) |
| 43 | \( 1 + 10.6T + 43T^{2} \) |
| 47 | \( 1 + (-5.49 + 9.52i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.271 + 0.470i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.16 + 7.21i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.963 + 1.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.50 + 6.07i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 + (2.58 + 4.47i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.57 - 6.19i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 + (1.60 - 2.78i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 13.7T + 97T^{2} \) |
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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
−9.195734532267681330516758877694, −8.523093384625663874464854984799, −7.59485892339912593784115614768, −6.90759432251702557101530662737, −6.31732980672441137087635173927, −5.03007923331713021583528870595, −4.69512222241321393648549188300, −3.50179592233065002228830580417, −2.44413649616669743096871901892, −1.65490205935222779477684947821,
0.42873652321206275857672553592, 1.44732781803402047033211546011, 2.91844887871293588357542268236, 3.84619079122321833208151679792, 4.49799975895806705351769628103, 5.51588363943881346190847740973, 6.23560333907585183435119924683, 7.31129429759146158012326601446, 7.69890130696843899323615025939, 8.687974760370452916251908221941