| L(s) = 1 | + (2.05 + 1.29i)2-s + (−2.93 + 1.02i)3-s + (0.827 + 1.71i)4-s + (5.87 − 1.34i)5-s + (−7.36 − 1.67i)6-s + (−9.36 − 4.50i)7-s + (0.569 − 5.05i)8-s + (0.506 − 0.404i)9-s + (13.8 + 4.83i)10-s + (0.977 + 8.67i)11-s + (−4.19 − 4.19i)12-s + (8.98 + 7.16i)13-s + (−13.4 − 21.3i)14-s + (−15.8 + 9.96i)15-s + (12.4 − 15.6i)16-s + (−9.77 + 9.77i)17-s + ⋯ |
| L(s) = 1 | + (1.02 + 0.646i)2-s + (−0.977 + 0.341i)3-s + (0.206 + 0.429i)4-s + (1.17 − 0.268i)5-s + (−1.22 − 0.279i)6-s + (−1.33 − 0.644i)7-s + (0.0711 − 0.631i)8-s + (0.0562 − 0.0448i)9-s + (1.38 + 0.483i)10-s + (0.0888 + 0.788i)11-s + (−0.349 − 0.349i)12-s + (0.691 + 0.551i)13-s + (−0.960 − 1.52i)14-s + (−1.05 + 0.664i)15-s + (0.779 − 0.976i)16-s + (−0.574 + 0.574i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.750 - 0.660i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.750 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.14785 + 0.433183i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.14785 + 0.433183i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + (28.9 + 1.48i)T \) |
| good | 2 | \( 1 + (-2.05 - 1.29i)T + (1.73 + 3.60i)T^{2} \) |
| 3 | \( 1 + (2.93 - 1.02i)T + (7.03 - 5.61i)T^{2} \) |
| 5 | \( 1 + (-5.87 + 1.34i)T + (22.5 - 10.8i)T^{2} \) |
| 7 | \( 1 + (9.36 + 4.50i)T + (30.5 + 38.3i)T^{2} \) |
| 11 | \( 1 + (-0.977 - 8.67i)T + (-117. + 26.9i)T^{2} \) |
| 13 | \( 1 + (-8.98 - 7.16i)T + (37.6 + 164. i)T^{2} \) |
| 17 | \( 1 + (9.77 - 9.77i)T - 289iT^{2} \) |
| 19 | \( 1 + (4.49 - 12.8i)T + (-282. - 225. i)T^{2} \) |
| 23 | \( 1 + (-6.44 + 28.2i)T + (-476. - 229. i)T^{2} \) |
| 31 | \( 1 + (-32.8 - 20.6i)T + (416. + 865. i)T^{2} \) |
| 37 | \( 1 + (-6.61 + 58.7i)T + (-1.33e3 - 304. i)T^{2} \) |
| 41 | \( 1 + (-28.8 - 28.8i)T + 1.68e3iT^{2} \) |
| 43 | \( 1 + (-23.3 - 37.1i)T + (-802. + 1.66e3i)T^{2} \) |
| 47 | \( 1 + (16.5 - 1.86i)T + (2.15e3 - 491. i)T^{2} \) |
| 53 | \( 1 + (12.8 + 56.5i)T + (-2.53e3 + 1.21e3i)T^{2} \) |
| 59 | \( 1 + 34.0T + 3.48e3T^{2} \) |
| 61 | \( 1 + (-5.90 + 2.06i)T + (2.90e3 - 2.32e3i)T^{2} \) |
| 67 | \( 1 + (53.1 - 42.4i)T + (998. - 4.37e3i)T^{2} \) |
| 71 | \( 1 + (24.4 + 19.5i)T + (1.12e3 + 4.91e3i)T^{2} \) |
| 73 | \( 1 + (43.4 - 27.2i)T + (2.31e3 - 4.80e3i)T^{2} \) |
| 79 | \( 1 + (-36.9 - 4.16i)T + (6.08e3 + 1.38e3i)T^{2} \) |
| 83 | \( 1 + (-62.9 + 30.2i)T + (4.29e3 - 5.38e3i)T^{2} \) |
| 89 | \( 1 + (-23.7 - 14.9i)T + (3.43e3 + 7.13e3i)T^{2} \) |
| 97 | \( 1 + (33.9 + 11.8i)T + (7.35e3 + 5.86e3i)T^{2} \) |
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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
−16.63370031734626087845225747455, −16.10085266833873541250810105564, −14.49314200630331247118845176446, −13.34056958368612604681149829939, −12.60384112745103212991983047438, −10.56836104612440131314766543871, −9.561785284806211974340524121393, −6.58527801647096413483558196111, −5.92273604664680627849655125928, −4.34963896828610810893406508030,
2.98020498858095359387189277077, 5.60425262649109687948534855117, 6.27690633875873339911594560134, 9.211208761656787579719812202007, 10.85578793440681775914991001985, 11.91015046454507410857762500919, 13.17666346820860989176358171566, 13.61312314174545842357691869043, 15.45853731049718275388510214040, 16.96850270222548521347984311408
