| L(s) = 1 | + (0.797 − 1.16i)2-s + (−0.381 − 0.606i)3-s + (−0.728 − 1.86i)4-s + (−0.388 − 0.309i)5-s + (−1.01 − 0.0384i)6-s + (1.45 + 0.331i)7-s + (−2.75 − 0.633i)8-s + (1.07 − 2.24i)9-s + (−0.671 + 0.206i)10-s + (−1.07 + 3.08i)11-s + (−0.852 + 1.15i)12-s + (0.610 + 1.26i)13-s + (1.54 − 1.43i)14-s + (−0.0398 + 0.353i)15-s + (−2.93 + 2.71i)16-s + (2.80 + 2.80i)17-s + ⋯ |
| L(s) = 1 | + (0.563 − 0.825i)2-s + (−0.220 − 0.350i)3-s + (−0.364 − 0.931i)4-s + (−0.173 − 0.138i)5-s + (−0.413 − 0.0156i)6-s + (0.548 + 0.125i)7-s + (−0.974 − 0.224i)8-s + (0.359 − 0.746i)9-s + (−0.212 + 0.0654i)10-s + (−0.325 + 0.929i)11-s + (−0.246 + 0.332i)12-s + (0.169 + 0.351i)13-s + (0.412 − 0.382i)14-s + (−0.0102 + 0.0913i)15-s + (−0.734 + 0.678i)16-s + (0.679 + 0.679i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0841 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0841 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.845630 - 0.920041i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.845630 - 0.920041i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.797 + 1.16i)T \) |
| 29 | \( 1 + (3.77 - 3.84i)T \) |
| good | 3 | \( 1 + (0.381 + 0.606i)T + (-1.30 + 2.70i)T^{2} \) |
| 5 | \( 1 + (0.388 + 0.309i)T + (1.11 + 4.87i)T^{2} \) |
| 7 | \( 1 + (-1.45 - 0.331i)T + (6.30 + 3.03i)T^{2} \) |
| 11 | \( 1 + (1.07 - 3.08i)T + (-8.60 - 6.85i)T^{2} \) |
| 13 | \( 1 + (-0.610 - 1.26i)T + (-8.10 + 10.1i)T^{2} \) |
| 17 | \( 1 + (-2.80 - 2.80i)T + 17iT^{2} \) |
| 19 | \( 1 + (-4.35 - 2.73i)T + (8.24 + 17.1i)T^{2} \) |
| 23 | \( 1 + (0.0143 - 0.0114i)T + (5.11 - 22.4i)T^{2} \) |
| 31 | \( 1 + (0.262 + 2.32i)T + (-30.2 + 6.89i)T^{2} \) |
| 37 | \( 1 + (1.69 - 0.592i)T + (28.9 - 23.0i)T^{2} \) |
| 41 | \( 1 + (0.378 - 0.378i)T - 41iT^{2} \) |
| 43 | \( 1 + (-3.80 - 0.429i)T + (41.9 + 9.56i)T^{2} \) |
| 47 | \( 1 + (8.34 + 2.92i)T + (36.7 + 29.3i)T^{2} \) |
| 53 | \( 1 + (6.44 - 8.07i)T + (-11.7 - 51.6i)T^{2} \) |
| 59 | \( 1 + 14.5iT - 59T^{2} \) |
| 61 | \( 1 + (-0.184 + 0.116i)T + (26.4 - 54.9i)T^{2} \) |
| 67 | \( 1 + (1.44 + 0.695i)T + (41.7 + 52.3i)T^{2} \) |
| 71 | \( 1 + (11.0 - 5.30i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (-6.61 - 0.745i)T + (71.1 + 16.2i)T^{2} \) |
| 79 | \( 1 + (-11.7 + 4.10i)T + (61.7 - 49.2i)T^{2} \) |
| 83 | \( 1 + (3.90 - 0.892i)T + (74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (-15.4 + 1.74i)T + (86.7 - 19.8i)T^{2} \) |
| 97 | \( 1 + (-8.35 - 5.25i)T + (42.0 + 87.3i)T^{2} \) |
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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
−12.94539502960845235697563774191, −12.26285187580630566386069505600, −11.53089146443454529818595690215, −10.25975786570214942029991825819, −9.343370316614218087692102717595, −7.79564009562298009327815204206, −6.31929455050718763705076084377, −5.01767141914849129237047327662, −3.68771325948842332926596240381, −1.64582911967373699507932559054,
3.29251063259574036230803361382, 4.84612261425193505724845920903, 5.68341852613240376109360278377, 7.34320484278456662336480571431, 8.056490131679543981913632881940, 9.460110004380808435085259580460, 10.93651000888375919131503813140, 11.72999928037819402202235135900, 13.18885358967916855501955408597, 13.80976786214340745944443210379
