L(s) = 1 | + (0.587 + 0.809i)9-s + (0.896 + 0.142i)13-s + (0.142 + 0.278i)17-s + (−1.11 + 0.363i)29-s + (−0.309 + 1.95i)37-s + (1.53 − 1.11i)41-s + i·49-s + (0.809 − 1.58i)53-s + (−1.53 − 1.11i)61-s + (0.278 + 1.76i)73-s + (−0.309 + 0.951i)81-s + (−0.690 + 0.951i)89-s + (1.76 + 0.896i)97-s + 1.61·101-s + (−0.363 − 0.5i)109-s + ⋯ |
L(s) = 1 | + (0.587 + 0.809i)9-s + (0.896 + 0.142i)13-s + (0.142 + 0.278i)17-s + (−1.11 + 0.363i)29-s + (−0.309 + 1.95i)37-s + (1.53 − 1.11i)41-s + i·49-s + (0.809 − 1.58i)53-s + (−1.53 − 1.11i)61-s + (0.278 + 1.76i)73-s + (−0.309 + 0.951i)81-s + (−0.690 + 0.951i)89-s + (1.76 + 0.896i)97-s + 1.61·101-s + (−0.363 − 0.5i)109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.396995945\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.396995945\) |
\(L(1)\) |
|
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 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.896 - 0.142i)T + (0.951 + 0.309i)T^{2} \) |
| 17 | \( 1 + (-0.142 - 0.278i)T + (-0.587 + 0.809i)T^{2} \) |
| 19 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 29 | \( 1 + (1.11 - 0.363i)T + (0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (0.309 - 1.95i)T + (-0.951 - 0.309i)T^{2} \) |
| 41 | \( 1 + (-1.53 + 1.11i)T + (0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (0.587 + 0.809i)T^{2} \) |
| 53 | \( 1 + (-0.809 + 1.58i)T + (-0.587 - 0.809i)T^{2} \) |
| 59 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (1.53 + 1.11i)T + (0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 71 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.278 - 1.76i)T + (-0.951 + 0.309i)T^{2} \) |
| 79 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 89 | \( 1 + (0.690 - 0.951i)T + (-0.309 - 0.951i)T^{2} \) |
| 97 | \( 1 + (-1.76 - 0.896i)T + (0.587 + 0.809i)T^{2} \) |
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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
−8.630649948431706275251795802346, −7.971341292194898766453443013427, −7.30129800685927133137385957413, −6.52257046765086949200433133824, −5.73192937273667745825583111487, −4.95707737001560491641149482377, −4.12947505690685488613356921454, −3.36066973990904861418209705415, −2.20348215493208934222310816228, −1.30743665669055765764464300510,
0.898831286554477018862416071550, 2.03401547895775351476235100903, 3.22599090325069779975953804756, 3.92421522720387031243609628335, 4.66020254663400576732024428217, 5.89740148548473940367181299954, 6.09434773798284084682200634270, 7.33213064179044085789625021096, 7.54853444452210774567697077109, 8.778567880969994457901475340473