L(s) = 1 | − 0.332i·2-s + i·3-s + 1.88·4-s + (−0.370 − 2.20i)5-s + 0.332·6-s − i·7-s − 1.29i·8-s − 9-s + (−0.733 + 0.123i)10-s − 11-s + 1.88i·12-s − 5.50i·13-s − 0.332·14-s + (2.20 − 0.370i)15-s + 3.34·16-s + 6.85i·17-s + ⋯ |
L(s) = 1 | − 0.235i·2-s + 0.577i·3-s + 0.944·4-s + (−0.165 − 0.986i)5-s + 0.135·6-s − 0.377i·7-s − 0.457i·8-s − 0.333·9-s + (−0.231 + 0.0389i)10-s − 0.301·11-s + 0.545i·12-s − 1.52i·13-s − 0.0888·14-s + (0.569 − 0.0957i)15-s + 0.837·16-s + 1.66i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.803327992\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.803327992\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 5 | \( 1 + (0.370 + 2.20i)T \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 + 0.332iT - 2T^{2} \) |
| 13 | \( 1 + 5.50iT - 13T^{2} \) |
| 17 | \( 1 - 6.85iT - 17T^{2} \) |
| 19 | \( 1 - 3.55T + 19T^{2} \) |
| 23 | \( 1 + 8.38iT - 23T^{2} \) |
| 29 | \( 1 + 8.24T + 29T^{2} \) |
| 31 | \( 1 - 0.581T + 31T^{2} \) |
| 37 | \( 1 + 9.80iT - 37T^{2} \) |
| 41 | \( 1 - 6.19T + 41T^{2} \) |
| 43 | \( 1 + 3.22iT - 43T^{2} \) |
| 47 | \( 1 + 6.73iT - 47T^{2} \) |
| 53 | \( 1 - 14.0iT - 53T^{2} \) |
| 59 | \( 1 + 2.05T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 - 1.52iT - 67T^{2} \) |
| 71 | \( 1 + 2.12T + 71T^{2} \) |
| 73 | \( 1 - 1.03iT - 73T^{2} \) |
| 79 | \( 1 - 1.77T + 79T^{2} \) |
| 83 | \( 1 - 4.43iT - 83T^{2} \) |
| 89 | \( 1 + 2.23T + 89T^{2} \) |
| 97 | \( 1 + 11.2iT - 97T^{2} \) |
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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
−9.806942764196632804321473996376, −8.754809278920652467124329874669, −7.996132144997069704195314088359, −7.33066765194179014867248517803, −5.93862664301412358058970931141, −5.49488174721335086641944957218, −4.21740052858116613309575538072, −3.45352131781296118392414957520, −2.20780897863573515660634990888, −0.76210947335356037926452585853,
1.69951702566412513874909434510, 2.64286515473304398819710000189, 3.48630796354956287045724748930, 5.10580722530882707477118630231, 6.00343144358689322194194881662, 6.84884814958717001768357996851, 7.33016341136069753686918759254, 7.926540508268924429759512015235, 9.295129131252319503714619823913, 9.843446574582917213464501496980