L(s) = 1 | + (0.258 − 0.965i)2-s + (−1.67 + 0.448i)3-s + (−0.866 − 0.499i)4-s + 1.73i·6-s + (−0.707 + 0.707i)8-s + (2.59 − 1.50i)9-s + (−1.5 + 0.866i)11-s + (1.67 + 0.448i)12-s + (−6.69 + 1.79i)13-s + (0.500 + 0.866i)16-s + (−2.12 − 2.12i)17-s + (−0.776 − 2.89i)18-s + 4i·19-s + (0.448 + 1.67i)22-s + (1.55 + 5.79i)23-s + (0.866 − 1.5i)24-s + ⋯ |
L(s) = 1 | + (0.183 − 0.683i)2-s + (−0.965 + 0.258i)3-s + (−0.433 − 0.249i)4-s + 0.707i·6-s + (−0.249 + 0.249i)8-s + (0.866 − 0.5i)9-s + (−0.452 + 0.261i)11-s + (0.482 + 0.129i)12-s + (−1.85 + 0.497i)13-s + (0.125 + 0.216i)16-s + (−0.514 − 0.514i)17-s + (−0.183 − 0.683i)18-s + 0.917i·19-s + (0.0955 + 0.356i)22-s + (0.323 + 1.20i)23-s + (0.176 − 0.306i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.300 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.180243 + 0.245884i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.180243 + 0.245884i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 + 0.965i)T \) |
| 3 | \( 1 + (1.67 - 0.448i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (1.5 - 0.866i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (6.69 - 1.79i)T + (11.2 - 6.5i)T^{2} \) |
| 17 | \( 1 + (2.12 + 2.12i)T + 17iT^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + (-1.55 - 5.79i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-1.73 - 3i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.89 - 4.89i)T - 37iT^{2} \) |
| 41 | \( 1 + (6 + 3.46i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.24 - 8.36i)T + (-37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + (-3.10 + 11.5i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-8.48 + 8.48i)T - 53iT^{2} \) |
| 59 | \( 1 + (4.33 - 7.5i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.896 + 3.34i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 3.46iT - 71T^{2} \) |
| 73 | \( 1 + (4.89 + 4.89i)T + 73iT^{2} \) |
| 79 | \( 1 + (3.46 - 2i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-8.69 - 2.32i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + 1.73T + 89T^{2} \) |
| 97 | \( 1 + (-15.0 - 4.03i)T + (84.0 + 48.5i)T^{2} \) |
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
−11.58676896728821843192415473383, −10.35830406397746661363402169428, −9.967614058129938504593253002180, −9.014035957040260557167061017312, −7.51098644418504124044281764924, −6.67554249959978129819931037939, −5.21487210689618369565076716029, −4.84996107510514202045537497253, −3.45065169928727962165776898290, −1.86935696868450519155710332313,
0.19224318159838953662073708944, 2.53896050480412485614984564258, 4.44111732952015781981362646042, 5.13025927054656271934021409533, 6.11703638023450213492472830376, 7.07133574963313552227755613380, 7.73048614041314370757164880935, 8.918346867630928656569936062450, 10.07390505741066109443630102132, 10.77335001856293753803851687013