L(s) = 1 | + (2.24 + 0.601i)2-s + (−0.173 − 1.72i)3-s + (2.93 + 1.69i)4-s + (0.647 − 3.97i)6-s + (0.201 − 0.751i)7-s + (2.29 + 2.29i)8-s + (−2.93 + 0.597i)9-s + (−0.220 + 0.127i)11-s + (2.41 − 5.36i)12-s + (0.992 + 3.70i)13-s + (0.903 − 1.56i)14-s + (0.367 + 0.636i)16-s + (−3.93 + 3.93i)17-s + (−6.95 − 0.427i)18-s + 0.440i·19-s + ⋯ |
L(s) = 1 | + (1.58 + 0.425i)2-s + (−0.100 − 0.994i)3-s + (1.46 + 0.848i)4-s + (0.264 − 1.62i)6-s + (0.0761 − 0.284i)7-s + (0.809 + 0.809i)8-s + (−0.979 + 0.199i)9-s + (−0.0663 + 0.0383i)11-s + (0.697 − 1.54i)12-s + (0.275 + 1.02i)13-s + (0.241 − 0.418i)14-s + (0.0918 + 0.159i)16-s + (−0.953 + 0.953i)17-s + (−1.63 − 0.100i)18-s + 0.101i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.177i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.177i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.53698 - 0.227210i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.53698 - 0.227210i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.173 + 1.72i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-2.24 - 0.601i)T + (1.73 + i)T^{2} \) |
| 7 | \( 1 + (-0.201 + 0.751i)T + (-6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (0.220 - 0.127i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.992 - 3.70i)T + (-11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 + (3.93 - 3.93i)T - 17iT^{2} \) |
| 19 | \( 1 - 0.440iT - 19T^{2} \) |
| 23 | \( 1 + (-3.42 + 0.917i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (2.76 + 4.78i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.0971 - 0.168i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.123 - 0.123i)T + 37iT^{2} \) |
| 41 | \( 1 + (3.88 + 2.24i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.33 + 0.357i)T + (37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (4.17 + 1.11i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (0.938 + 0.938i)T + 53iT^{2} \) |
| 59 | \( 1 + (4.02 - 6.96i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.44 + 2.49i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-12.9 + 3.47i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 2.15iT - 71T^{2} \) |
| 73 | \( 1 + (-9.18 + 9.18i)T - 73iT^{2} \) |
| 79 | \( 1 + (-11.9 + 6.91i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.39 + 5.20i)T + (-71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 - 0.285T + 89T^{2} \) |
| 97 | \( 1 + (-2.34 + 8.73i)T + (-84.0 - 48.5i)T^{2} \) |
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
−12.50369119048513052794665187952, −11.66701740791377203081157634270, −10.86176691015259898725400457281, −9.018725320015374766187883920744, −7.77318618744645989959144349369, −6.73632325418328369412375932111, −6.17008646090655856619332606485, −4.89955566838860141138499768875, −3.71891586839561167832284671242, −2.13387744146470272935746528066,
2.68455126968871629347101327339, 3.66187296808383412624866064450, 4.91186929975716370112056249518, 5.47435673431666687160071722471, 6.72654755633397998272410626495, 8.474613704917700239084984779660, 9.607969259126201067422355791739, 10.86996534071823485425051000403, 11.27221405188945832041693867710, 12.34172991354759908267307930874