L(s) = 1 | + (−1.15 + 1.99i)2-s + (0.5 − 0.866i)3-s + (−1.65 − 2.86i)4-s + (1.15 + 1.99i)6-s + (−0.5 − 0.866i)7-s + 2.99·8-s + (1 + 1.73i)9-s + (−0.802 + 1.39i)11-s − 3.30·12-s + 3.60·13-s + 2.30·14-s + (−0.151 + 0.262i)16-s + (3.80 + 6.58i)17-s − 4.60·18-s + (2.80 + 4.85i)19-s + ⋯ |
L(s) = 1 | + (−0.814 + 1.41i)2-s + (0.288 − 0.499i)3-s + (−0.825 − 1.43i)4-s + (0.470 + 0.814i)6-s + (−0.188 − 0.327i)7-s + 1.06·8-s + (0.333 + 0.577i)9-s + (−0.242 + 0.419i)11-s − 0.953·12-s + 1.00·13-s + 0.615·14-s + (−0.0378 + 0.0655i)16-s + (0.922 + 1.59i)17-s − 1.08·18-s + (0.643 + 1.11i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.565411 + 0.731987i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.565411 + 0.731987i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 13 | \( 1 - 3.60T \) |
good | 2 | \( 1 + (1.15 - 1.99i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.5 + 0.866i)T + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.802 - 1.39i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-3.80 - 6.58i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.80 - 4.85i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.10 + 5.37i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + (-1.80 + 3.12i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5.10 + 8.84i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 9.21T + 47T^{2} \) |
| 53 | \( 1 - 3.21T + 53T^{2} \) |
| 59 | \( 1 + (-5.40 - 9.36i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.5 - 6.06i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (2.40 + 4.17i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 0.788T + 73T^{2} \) |
| 79 | \( 1 - 5.21T + 79T^{2} \) |
| 83 | \( 1 - 9.21T + 83T^{2} \) |
| 89 | \( 1 + (-3.10 + 5.37i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (4.19 + 7.26i)T + (-48.5 + 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.96846106842072110437355093054, −10.39085008000391255936837556074, −9.961303153497257955958289818000, −8.597627009926907590430072397338, −7.972606871790070414945549898896, −7.33900165743966071498081908284, −6.26732699411462469006294834761, −5.44493730143086459700876841250, −3.76875283990016121837267682619, −1.52332187916338461010133311046,
0.977804594119646731272313460937, 2.85810061674438127217345055954, 3.50328552729800497329354268140, 5.00229982813411728945469653823, 6.59410422657347910222191309214, 8.034059161771232588227221268947, 9.037059217912272236157611903365, 9.481161302937695425632717230012, 10.35326202551668833516751277073, 11.26087904461792788765892937685