L(s) = 1 | + (−0.5 + 0.866i)5-s + (0.194 − 2.63i)7-s + (1.26 + 2.19i)11-s + 5.45·13-s + (−3.25 − 5.64i)17-s + (−0.0406 + 0.0703i)19-s + (1.23 − 2.13i)23-s + (−0.499 − 0.866i)25-s − 8.37·29-s + (−0.852 − 1.47i)31-s + (2.18 + 1.48i)35-s + (1.18 − 2.05i)37-s + 3.75·41-s + 9.44·43-s + (−0.962 + 1.66i)47-s + ⋯ |
L(s) = 1 | + (−0.223 + 0.387i)5-s + (0.0736 − 0.997i)7-s + (0.381 + 0.660i)11-s + 1.51·13-s + (−0.790 − 1.36i)17-s + (−0.00932 + 0.0161i)19-s + (0.257 − 0.446i)23-s + (−0.0999 − 0.173i)25-s − 1.55·29-s + (−0.153 − 0.265i)31-s + (0.369 + 0.251i)35-s + (0.195 − 0.338i)37-s + 0.585·41-s + 1.43·43-s + (−0.140 + 0.243i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.491 + 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.491 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.658946832\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.658946832\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.194 + 2.63i)T \) |
good | 11 | \( 1 + (-1.26 - 2.19i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 5.45T + 13T^{2} \) |
| 17 | \( 1 + (3.25 + 5.64i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.0406 - 0.0703i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.23 + 2.13i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 8.37T + 29T^{2} \) |
| 31 | \( 1 + (0.852 + 1.47i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.18 + 2.05i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 3.75T + 41T^{2} \) |
| 43 | \( 1 - 9.44T + 43T^{2} \) |
| 47 | \( 1 + (0.962 - 1.66i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.49 - 4.31i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.65 - 2.87i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.150 - 0.261i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.58 + 2.74i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 0.684T + 71T^{2} \) |
| 73 | \( 1 + (7.64 + 13.2i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.29 + 12.6i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 12.0T + 83T^{2} \) |
| 89 | \( 1 + (-1.79 + 3.11i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 9.20T + 97T^{2} \) |
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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
−8.996333402153057448712762428265, −7.77029059248640031633054472703, −7.32118652010166991156616787831, −6.58620640693450718772769544425, −5.79632807488559338624999381221, −4.56810905576349407113827353663, −4.03792908609268939200788428587, −3.12567305400392265436991511571, −1.89239289817448895109496838354, −0.62127155595022370120142160993,
1.19014659850874504236755338937, 2.21420759797314944021278863539, 3.54050418900726826133735881568, 4.04599212254946115777426101459, 5.30282227975659945115247174745, 5.94783787380613938029060735939, 6.50087122452688662621350660141, 7.70251709156242053335412948017, 8.518178189004584881024891266335, 8.836542193035612241223192867719