L(s) = 1 | + (0.866 − 0.5i)5-s + (−0.642 + 0.766i)9-s + (−0.766 + 0.642i)13-s + (1.26 − 1.50i)17-s + (0.499 − 0.866i)25-s + (1.92 − 0.515i)29-s + (0.642 + 0.766i)37-s + (1.26 + 1.50i)41-s + (−0.173 + 0.984i)45-s + (−0.342 − 0.939i)49-s + (0.296 + 0.424i)53-s + (0.0151 − 0.173i)61-s + (−0.342 + 0.939i)65-s + (−1.36 − 1.36i)73-s + (−0.173 − 0.984i)81-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)5-s + (−0.642 + 0.766i)9-s + (−0.766 + 0.642i)13-s + (1.26 − 1.50i)17-s + (0.499 − 0.866i)25-s + (1.92 − 0.515i)29-s + (0.642 + 0.766i)37-s + (1.26 + 1.50i)41-s + (−0.173 + 0.984i)45-s + (−0.342 − 0.939i)49-s + (0.296 + 0.424i)53-s + (0.0151 − 0.173i)61-s + (−0.342 + 0.939i)65-s + (−1.36 − 1.36i)73-s + (−0.173 − 0.984i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.421421126\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.421421126\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.866 + 0.5i)T \) |
| 37 | \( 1 + (-0.642 - 0.766i)T \) |
good | 3 | \( 1 + (0.642 - 0.766i)T^{2} \) |
| 7 | \( 1 + (0.342 + 0.939i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \) |
| 17 | \( 1 + (-1.26 + 1.50i)T + (-0.173 - 0.984i)T^{2} \) |
| 19 | \( 1 + (-0.642 + 0.766i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-1.92 + 0.515i)T + (0.866 - 0.5i)T^{2} \) |
| 31 | \( 1 + iT^{2} \) |
| 41 | \( 1 + (-1.26 - 1.50i)T + (-0.173 + 0.984i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (-0.296 - 0.424i)T + (-0.342 + 0.939i)T^{2} \) |
| 59 | \( 1 + (0.342 - 0.939i)T^{2} \) |
| 61 | \( 1 + (-0.0151 + 0.173i)T + (-0.984 - 0.173i)T^{2} \) |
| 67 | \( 1 + (-0.342 - 0.939i)T^{2} \) |
| 71 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 73 | \( 1 + (1.36 + 1.36i)T + iT^{2} \) |
| 79 | \( 1 + (-0.342 - 0.939i)T^{2} \) |
| 83 | \( 1 + (-0.984 + 0.173i)T^{2} \) |
| 89 | \( 1 + (0.939 - 0.657i)T + (0.342 - 0.939i)T^{2} \) |
| 97 | \( 1 + (-1.32 + 0.766i)T + (0.5 - 0.866i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.956471574121515883819159663764, −8.155175924261900120968792433635, −7.51171600939045450740000132282, −6.54196576952749463480933207926, −5.79935526828650828235390577036, −4.94199575838138056118448079116, −4.59295325119970380945233329963, −2.94718704036420210746721681196, −2.43172829774818466656617828423, −1.11719719782028203895762089515,
1.18853843382571882587731570241, 2.51811238085617191230398127431, 3.16505818144193450568003275732, 4.15394570695657512492783111148, 5.42600551944500450858754232164, 5.83496874327769351457738492102, 6.56523502417828964114743109496, 7.42427101585761370232886465153, 8.242796893098948015996589739857, 8.979010416372772439434936855884