L(s) = 1 | + (−0.415 + 0.719i)2-s + (0.786 + 1.36i)3-s + (0.154 + 0.268i)4-s − 1.30·6-s + (−0.888 + 0.458i)7-s − 1.08·8-s + (−0.735 + 1.27i)9-s + (0.327 + 0.566i)11-s + (−0.243 + 0.421i)12-s + (0.0395 − 0.829i)14-s + (0.297 − 0.514i)16-s + (−0.611 − 1.05i)18-s + (0.959 − 1.66i)19-s + (−1.32 − 0.849i)21-s − 0.543·22-s + ⋯ |
L(s) = 1 | + (−0.415 + 0.719i)2-s + (0.786 + 1.36i)3-s + (0.154 + 0.268i)4-s − 1.30·6-s + (−0.888 + 0.458i)7-s − 1.08·8-s + (−0.735 + 1.27i)9-s + (0.327 + 0.566i)11-s + (−0.243 + 0.421i)12-s + (0.0395 − 0.829i)14-s + (0.297 − 0.514i)16-s + (−0.611 − 1.05i)18-s + (0.959 − 1.66i)19-s + (−1.32 − 0.849i)21-s − 0.543·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0157i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0157i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.003902603\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.003902603\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.888 - 0.458i)T \) |
| 167 | \( 1 - T \) |
good | 2 | \( 1 + (0.415 - 0.719i)T + (-0.5 - 0.866i)T^{2} \) |
| 3 | \( 1 + (-0.786 - 1.36i)T + (-0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.327 - 0.566i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.959 + 1.66i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - 0.0951T + T^{2} \) |
| 31 | \( 1 + (-0.995 - 1.72i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.723 - 1.25i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.142 + 0.246i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.928 - 1.60i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + 1.30T + T^{2} \) |
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
−9.859053440746531218364254986430, −9.508265653831072967171768323230, −8.818323946873001631209362364986, −8.200187222278190142717399865945, −7.08680591736172281566771059643, −6.44504455673886402242955739378, −5.24560871857655871030266876828, −4.31115095633105478323091106232, −3.17724832855386679931568926252, −2.70381767145151592137610108266,
0.957728265793958044145439233526, 1.94364842905917029471732187292, 3.02912358110247456125282205765, 3.71248774068331268112267783090, 5.75143806025927618542818174562, 6.30351294389630838880113759962, 7.20455477193480249974250510200, 7.961239443549511271353979477248, 8.787468493522879848892038775444, 9.706235673531725141674665417591