L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + 3·5-s + (2.5 − 0.866i)7-s + 0.999·8-s + (−1.5 − 2.59i)10-s + (2 + 3.46i)13-s + (−2 − 1.73i)14-s + (−0.5 − 0.866i)16-s + (−3 − 5.19i)17-s + (2 − 3.46i)19-s + (−1.49 + 2.59i)20-s + 6·23-s + 4·25-s + (1.99 − 3.46i)26-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + 1.34·5-s + (0.944 − 0.327i)7-s + 0.353·8-s + (−0.474 − 0.821i)10-s + (0.554 + 0.960i)13-s + (−0.534 − 0.462i)14-s + (−0.125 − 0.216i)16-s + (−0.727 − 1.26i)17-s + (0.458 − 0.794i)19-s + (−0.335 + 0.580i)20-s + 1.25·23-s + 0.800·25-s + (0.392 − 0.679i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 + 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.678 + 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.934749980\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.934749980\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.5 + 0.866i)T \) |
good | 5 | \( 1 - 3T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + (-2 - 3.46i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4 - 6.92i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3 - 5.19i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4 - 6.92i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.5 + 7.79i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5 + 8.66i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (8.5 + 14.7i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-6 + 10.3i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5 + 8.66i)T + (-48.5 - 84.0i)T^{2} \) |
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
−9.640030554195219622724453980415, −9.084472269633504340424157681700, −8.383873441905136549870306355910, −7.12331760036444346428040246791, −6.54640407873483159454704105046, −5.11602555667217893275608946628, −4.70378989197891396018085046578, −3.17529981562075934896017677525, −2.09347000800591902414968162116, −1.20216782760630861538981347015,
1.33321183168770745970890973673, 2.27832203243311613853712971546, 3.86831554004392299866370909266, 5.27372505804925918535924017969, 5.62353669179929321937744103968, 6.45796088565021381524879969472, 7.52730299436907389177009646748, 8.382789538456253496894776650982, 8.982693336642479336174984220168, 9.796041074028343769302268254113