L(s) = 1 | + (1.14 − 0.828i)2-s + 0.692i·3-s + (0.627 − 1.89i)4-s + (0.573 + 0.794i)6-s + (0.343 + 0.343i)7-s + (−0.853 − 2.69i)8-s + 2.52·9-s + (0.843 + 0.843i)11-s + (1.31 + 0.434i)12-s + 3.68·13-s + (0.678 + 0.109i)14-s + (−3.21 − 2.38i)16-s + (−0.412 − 0.412i)17-s + (2.88 − 2.08i)18-s + (−5.37 − 5.37i)19-s + ⋯ |
L(s) = 1 | + (0.810 − 0.585i)2-s + 0.399i·3-s + (0.313 − 0.949i)4-s + (0.234 + 0.324i)6-s + (0.129 + 0.129i)7-s + (−0.301 − 0.953i)8-s + 0.840·9-s + (0.254 + 0.254i)11-s + (0.379 + 0.125i)12-s + 1.02·13-s + (0.181 + 0.0292i)14-s + (−0.802 − 0.596i)16-s + (−0.0999 − 0.0999i)17-s + (0.680 − 0.492i)18-s + (−1.23 − 1.23i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.697 + 0.717i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.697 + 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.08800 - 0.882280i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.08800 - 0.882280i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.14 + 0.828i)T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 0.692iT - 3T^{2} \) |
| 7 | \( 1 + (-0.343 - 0.343i)T + 7iT^{2} \) |
| 11 | \( 1 + (-0.843 - 0.843i)T + 11iT^{2} \) |
| 13 | \( 1 - 3.68T + 13T^{2} \) |
| 17 | \( 1 + (0.412 + 0.412i)T + 17iT^{2} \) |
| 19 | \( 1 + (5.37 + 5.37i)T + 19iT^{2} \) |
| 23 | \( 1 + (-3.08 + 3.08i)T - 23iT^{2} \) |
| 29 | \( 1 + (4.22 - 4.22i)T - 29iT^{2} \) |
| 31 | \( 1 - 8.75iT - 31T^{2} \) |
| 37 | \( 1 - 5.41T + 37T^{2} \) |
| 41 | \( 1 - 2.54iT - 41T^{2} \) |
| 43 | \( 1 + 4.30T + 43T^{2} \) |
| 47 | \( 1 + (4.56 - 4.56i)T - 47iT^{2} \) |
| 53 | \( 1 - 6.07iT - 53T^{2} \) |
| 59 | \( 1 + (7.33 - 7.33i)T - 59iT^{2} \) |
| 61 | \( 1 + (4.81 + 4.81i)T + 61iT^{2} \) |
| 67 | \( 1 + 14.3T + 67T^{2} \) |
| 71 | \( 1 + 2.97T + 71T^{2} \) |
| 73 | \( 1 + (-6.87 - 6.87i)T + 73iT^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 + 7.15iT - 83T^{2} \) |
| 89 | \( 1 - 1.10T + 89T^{2} \) |
| 97 | \( 1 + (7.15 + 7.15i)T + 97iT^{2} \) |
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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
−10.90898672925254704273442584197, −10.71053099227269201836818290038, −9.469879976754019995299173021316, −8.720986923120419194974922352030, −7.06617767371384588563319309122, −6.31705533561526512040458184591, −4.94791511119656416017461162951, −4.27687578344479294420174243021, −3.07526202527956961194773940195, −1.52638577605510697770055692955,
1.86317585544169625288127150032, 3.62642719331490457294038521952, 4.40302375629971769227412842078, 5.87390017324263360262196023889, 6.46702354514011624934411617481, 7.61259534689029721072712882917, 8.238302908411195039044024368586, 9.439403528655293645670432651592, 10.75181215515465241266024934632, 11.55449713437351075213646258809