L(s) = 1 | + (−0.122 + 0.350i)2-s + (0.563 + 0.826i)3-s + (0.673 + 0.537i)4-s + (−0.359 + 0.0962i)6-s + (−0.586 + 0.368i)8-s + (−0.365 + 0.930i)9-s + (−0.0643 + 0.859i)12-s + (1.61 − 0.367i)13-s + (0.134 + 0.589i)16-s + (−0.281 − 0.242i)18-s + (0.433 − 0.900i)23-s + (−0.634 − 0.276i)24-s + (0.623 − 0.781i)25-s + (−0.0687 + 0.610i)26-s + (−0.974 + 0.222i)27-s + ⋯ |
L(s) = 1 | + (−0.122 + 0.350i)2-s + (0.563 + 0.826i)3-s + (0.673 + 0.537i)4-s + (−0.359 + 0.0962i)6-s + (−0.586 + 0.368i)8-s + (−0.365 + 0.930i)9-s + (−0.0643 + 0.859i)12-s + (1.61 − 0.367i)13-s + (0.134 + 0.589i)16-s + (−0.281 − 0.242i)18-s + (0.433 − 0.900i)23-s + (−0.634 − 0.276i)24-s + (0.623 − 0.781i)25-s + (−0.0687 + 0.610i)26-s + (−0.974 + 0.222i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.253 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.253 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.568522378\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.568522378\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.563 - 0.826i)T \) |
| 23 | \( 1 + (-0.433 + 0.900i)T \) |
| 29 | \( 1 + (0.149 - 0.988i)T \) |
good | 2 | \( 1 + (0.122 - 0.350i)T + (-0.781 - 0.623i)T^{2} \) |
| 5 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 7 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 11 | \( 1 + (-0.433 - 0.900i)T^{2} \) |
| 13 | \( 1 + (-1.61 + 0.367i)T + (0.900 - 0.433i)T^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + (-0.974 + 0.222i)T^{2} \) |
| 31 | \( 1 + (1.82 + 0.638i)T + (0.781 + 0.623i)T^{2} \) |
| 37 | \( 1 + (0.433 - 0.900i)T^{2} \) |
| 41 | \( 1 + (0.660 + 0.660i)T + iT^{2} \) |
| 43 | \( 1 + (0.781 - 0.623i)T^{2} \) |
| 47 | \( 1 + (0.631 - 1.00i)T + (-0.433 - 0.900i)T^{2} \) |
| 53 | \( 1 + (0.623 - 0.781i)T^{2} \) |
| 59 | \( 1 + 1.24iT - T^{2} \) |
| 61 | \( 1 + (0.974 + 0.222i)T^{2} \) |
| 67 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 71 | \( 1 + (0.443 + 1.94i)T + (-0.900 + 0.433i)T^{2} \) |
| 73 | \( 1 + (0.754 - 0.264i)T + (0.781 - 0.623i)T^{2} \) |
| 79 | \( 1 + (-0.433 + 0.900i)T^{2} \) |
| 83 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 89 | \( 1 + (0.781 + 0.623i)T^{2} \) |
| 97 | \( 1 + (0.974 - 0.222i)T^{2} \) |
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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
−9.265728928815717303201003324057, −8.733819150351100092167918922407, −8.156604985971034291425257665309, −7.37454327767040735407883088949, −6.42759137449149729654532132907, −5.69986597122469064861999683863, −4.63690124164868306548319173062, −3.58377275895689628562002154963, −3.06869809704114481014248790347, −1.87695689182312941589480882495,
1.24253666255304944698498926373, 1.90931047569524651794644609896, 3.13500791407756157716118916220, 3.74882319212210966561247769035, 5.38469476780045361656433651421, 6.07024260998943330155283762417, 6.87589917294847579488011123227, 7.39509890971605313835595977606, 8.503908179704598716645229706957, 9.022760834183275995906745163276