L(s) = 1 | + i·5-s + (−1 + i)13-s + (1 + i)17-s − 25-s + (1 + i)37-s − i·49-s + (−1 + i)53-s + (−1 − i)65-s + (1 − i)73-s + (−1 + i)85-s + (1 + i)97-s + 2·101-s − 2i·109-s + (1 − i)113-s + ⋯ |
L(s) = 1 | + i·5-s + (−1 + i)13-s + (1 + i)17-s − 25-s + (1 + i)37-s − i·49-s + (−1 + i)53-s + (−1 − i)65-s + (1 − i)73-s + (−1 + i)85-s + (1 + i)97-s + 2·101-s − 2i·109-s + (1 − i)113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.003579579\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.003579579\) |
\(L(1)\) |
|
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 | \( 1 - iT \) |
good | 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (1 - i)T - iT^{2} \) |
| 17 | \( 1 + (-1 - i)T + iT^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (-1 - i)T + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (1 - i)T - iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-1 + i)T - iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-1 - i)T + iT^{2} \) |
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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
−9.948291209855228656984027196396, −9.268509675723928818072588730068, −8.120543968661658975165355533118, −7.47779700011647836387520347388, −6.63557196822911011547672148525, −5.96945497089711161558334807659, −4.83510087779419007068133314720, −3.83987950428104676100348833714, −2.88015613929687152014350821595, −1.80431104230119115090236852145,
0.848296586500977528890620058934, 2.38113606625636665596598092352, 3.48938460250877756966617029190, 4.70980916418435184764549695298, 5.26430892208687826208643730303, 6.10480684256356854509112507007, 7.45672434476210826111335352555, 7.81194292611428217007354855222, 8.805170338517886325761611565369, 9.619135127605727888345857042665