L(s) = 1 | + (−0.951 + 0.309i)9-s + (−0.278 − 0.142i)13-s + (−1.76 − 0.278i)17-s + (−1.11 + 1.53i)29-s + (−0.412 + 0.809i)37-s + (0.363 + 1.11i)41-s − i·49-s + (−1.95 + 0.309i)53-s + (−0.363 + 1.11i)61-s + (−0.896 − 1.76i)73-s + (0.809 − 0.587i)81-s + (1.80 + 0.587i)89-s + (−0.142 − 0.896i)97-s − 0.618·101-s + (−1.53 + 0.5i)109-s + ⋯ |
L(s) = 1 | + (−0.951 + 0.309i)9-s + (−0.278 − 0.142i)13-s + (−1.76 − 0.278i)17-s + (−1.11 + 1.53i)29-s + (−0.412 + 0.809i)37-s + (0.363 + 1.11i)41-s − i·49-s + (−1.95 + 0.309i)53-s + (−0.363 + 1.11i)61-s + (−0.896 − 1.76i)73-s + (0.809 − 0.587i)81-s + (1.80 + 0.587i)89-s + (−0.142 − 0.896i)97-s − 0.618·101-s + (−1.53 + 0.5i)109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2853206794\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2853206794\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (0.278 + 0.142i)T + (0.587 + 0.809i)T^{2} \) |
| 17 | \( 1 + (1.76 + 0.278i)T + (0.951 + 0.309i)T^{2} \) |
| 19 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 29 | \( 1 + (1.11 - 1.53i)T + (-0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (0.412 - 0.809i)T + (-0.587 - 0.809i)T^{2} \) |
| 41 | \( 1 + (-0.363 - 1.11i)T + (-0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 53 | \( 1 + (1.95 - 0.309i)T + (0.951 - 0.309i)T^{2} \) |
| 59 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (0.363 - 1.11i)T + (-0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 71 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (0.896 + 1.76i)T + (-0.587 + 0.809i)T^{2} \) |
| 79 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 89 | \( 1 + (-1.80 - 0.587i)T + (0.809 + 0.587i)T^{2} \) |
| 97 | \( 1 + (0.142 + 0.896i)T + (-0.951 + 0.309i)T^{2} \) |
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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
−8.962016502698125478465909867741, −8.271039984676221234090137167219, −7.49702533751218186170909572541, −6.71169160202108214149234666725, −6.04152948710413406921592244264, −5.11091042815852925073194855450, −4.58873938611147213913193441051, −3.42625349440374281574296387389, −2.66769380532270550784370097460, −1.69603992315337827643740621500,
0.14373568290828362381687220266, 1.93138120635499229201319970850, 2.66886965184114505833188087916, 3.75812110907431574083513251902, 4.45094088031487604052914311015, 5.41848009467673031714237459598, 6.13225850348543145384816549660, 6.75136905827156665914065291522, 7.66303704782819474545662029096, 8.322610863824198569568538997803