L(s) = 1 | + (0.866 − 0.5i)3-s + (−0.866 − 0.5i)7-s + (0.499 − 0.866i)9-s + 13-s + (0.866 + 0.5i)19-s − 0.999·21-s + (−0.5 − 0.866i)25-s − 0.999i·27-s + (−0.866 − 1.5i)31-s + (1.5 + 0.866i)37-s + (0.866 − 0.5i)39-s − 1.73·43-s + (0.499 + 0.866i)49-s + 0.999·57-s + (−1 + 1.73i)61-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)3-s + (−0.866 − 0.5i)7-s + (0.499 − 0.866i)9-s + 13-s + (0.866 + 0.5i)19-s − 0.999·21-s + (−0.5 − 0.866i)25-s − 0.999i·27-s + (−0.866 − 1.5i)31-s + (1.5 + 0.866i)37-s + (0.866 − 0.5i)39-s − 1.73·43-s + (0.499 + 0.866i)49-s + 0.999·57-s + (−1 + 1.73i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.370328183\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.370328183\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 7 | \( 1 + (0.866 + 0.5i)T \) |
good | 5 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + 1.73T + T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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
−9.731650140802789143387061763502, −8.839124653420185346405952029439, −8.035401235356855254279873524931, −7.36636436887783695457327094499, −6.45736934436914477410932153742, −5.82390442804587029414202348613, −4.21700175689204017603236689599, −3.55191301380324080958856553776, −2.59522956927817490341761031319, −1.20369498677818384667231625106,
1.79216706521558230523387072872, 3.14331303350506347987763200447, 3.54567093634964191851873930560, 4.81668943185057799191568386224, 5.71852831274457910056582509979, 6.71438639484609112962833061995, 7.60946803606171275978168152518, 8.481207374515345013995452309078, 9.258451704606328313511553112093, 9.601843975137245042880584311686