L(s) = 1 | + i·2-s − 4-s + (−1.22 + 2.12i)5-s + (2.62 + 0.358i)7-s − i·8-s + (−2.12 − 1.22i)10-s + (3.67 − 2.12i)11-s + (3.62 − 2.09i)13-s + (−0.358 + 2.62i)14-s + 16-s + (−1.22 + 2.12i)17-s + (4.24 − 2.44i)19-s + (1.22 − 2.12i)20-s + (2.12 + 3.67i)22-s + (−5.19 − 3i)23-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (−0.547 + 0.948i)5-s + (0.990 + 0.135i)7-s − 0.353i·8-s + (−0.670 − 0.387i)10-s + (1.10 − 0.639i)11-s + (1.00 − 0.579i)13-s + (−0.0958 + 0.700i)14-s + 0.250·16-s + (−0.297 + 0.514i)17-s + (0.973 − 0.561i)19-s + (0.273 − 0.474i)20-s + (0.452 + 0.783i)22-s + (−1.08 − 0.625i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.182 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.182 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.744589960\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.744589960\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.62 - 0.358i)T \) |
good | 5 | \( 1 + (1.22 - 2.12i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-3.67 + 2.12i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.62 + 2.09i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.22 - 2.12i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.24 + 2.44i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (5.19 + 3i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-8.87 - 5.12i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 5.61iT - 31T^{2} \) |
| 37 | \( 1 + (-1.62 - 2.80i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.22 - 2.12i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.5 - 6.06i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 7.94T + 47T^{2} \) |
| 53 | \( 1 + (-2.15 - 1.24i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 2.44T + 59T^{2} \) |
| 61 | \( 1 + 0.717iT - 61T^{2} \) |
| 67 | \( 1 + 3.48T + 67T^{2} \) |
| 71 | \( 1 - 12.7iT - 71T^{2} \) |
| 73 | \( 1 + (13.2 + 7.64i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 9.24T + 79T^{2} \) |
| 83 | \( 1 + (-2.74 + 4.75i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-8.87 - 15.3i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.74 - 3.31i)T + (48.5 + 84.0i)T^{2} \) |
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
−10.00531914463315031674899018377, −8.836675729762017274527227630906, −8.275231359462432122370677456113, −7.58845340620133918063153971843, −6.54450787831757902538090400586, −6.07682907449036740123607053311, −4.85582945185529750376317228279, −3.90822630776541845787388495475, −2.99814123435752513156012241553, −1.19693929740338588308961457731,
1.04212164399707012814132861656, 1.86454365711732429432244125663, 3.59216824933628714410565689020, 4.35181920288935479426975811294, 4.95525597123336026128138993930, 6.17369462702358746406174881964, 7.35527872169449936870460906636, 8.268437095208625338282583301688, 8.797368642322222274675612726605, 9.609320006720465231776192281942