| L(s) = 1 | + (−0.996 − 1.00i)2-s + (0.0826 + 0.733i)3-s + (−0.0151 + 1.99i)4-s + (0.524 − 1.08i)5-s + (0.653 − 0.813i)6-s + (2.09 + 1.67i)7-s + (2.02 − 1.97i)8-s + (2.39 − 0.546i)9-s + (−1.61 + 0.558i)10-s + (−0.513 − 0.816i)11-s + (−1.46 + 0.154i)12-s + (0.813 + 0.185i)13-s + (−0.410 − 3.76i)14-s + (0.842 + 0.294i)15-s + (−3.99 − 0.0607i)16-s + (0.315 + 0.315i)17-s + ⋯ |
| L(s) = 1 | + (−0.704 − 0.709i)2-s + (0.0477 + 0.423i)3-s + (−0.00759 + 0.999i)4-s + (0.234 − 0.487i)5-s + (0.266 − 0.332i)6-s + (0.792 + 0.631i)7-s + (0.715 − 0.699i)8-s + (0.797 − 0.182i)9-s + (−0.510 + 0.176i)10-s + (−0.154 − 0.246i)11-s + (−0.423 + 0.0444i)12-s + (0.225 + 0.0514i)13-s + (−0.109 − 1.00i)14-s + (0.217 + 0.0760i)15-s + (−0.999 − 0.0151i)16-s + (0.0765 + 0.0765i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 + 0.359i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.933 + 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.849271 - 0.157749i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.849271 - 0.157749i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.996 + 1.00i)T \) |
| 29 | \( 1 + (5.33 - 0.756i)T \) |
| good | 3 | \( 1 + (-0.0826 - 0.733i)T + (-2.92 + 0.667i)T^{2} \) |
| 5 | \( 1 + (-0.524 + 1.08i)T + (-3.11 - 3.90i)T^{2} \) |
| 7 | \( 1 + (-2.09 - 1.67i)T + (1.55 + 6.82i)T^{2} \) |
| 11 | \( 1 + (0.513 + 0.816i)T + (-4.77 + 9.91i)T^{2} \) |
| 13 | \( 1 + (-0.813 - 0.185i)T + (11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (-0.315 - 0.315i)T + 17iT^{2} \) |
| 19 | \( 1 + (2.86 + 0.323i)T + (18.5 + 4.22i)T^{2} \) |
| 23 | \( 1 + (1.25 + 2.61i)T + (-14.3 + 17.9i)T^{2} \) |
| 31 | \( 1 + (9.65 - 3.37i)T + (24.2 - 19.3i)T^{2} \) |
| 37 | \( 1 + (2.38 + 1.49i)T + (16.0 + 33.3i)T^{2} \) |
| 41 | \( 1 + (2.18 - 2.18i)T - 41iT^{2} \) |
| 43 | \( 1 + (-3.84 + 10.9i)T + (-33.6 - 26.8i)T^{2} \) |
| 47 | \( 1 + (-7.43 + 4.66i)T + (20.3 - 42.3i)T^{2} \) |
| 53 | \( 1 + (9.37 + 4.51i)T + (33.0 + 41.4i)T^{2} \) |
| 59 | \( 1 + 5.67iT - 59T^{2} \) |
| 61 | \( 1 + (2.12 - 0.239i)T + (59.4 - 13.5i)T^{2} \) |
| 67 | \( 1 + (-2.93 - 12.8i)T + (-60.3 + 29.0i)T^{2} \) |
| 71 | \( 1 + (0.849 - 3.72i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (1.93 - 5.51i)T + (-57.0 - 45.5i)T^{2} \) |
| 79 | \( 1 + (8.75 + 5.49i)T + (34.2 + 71.1i)T^{2} \) |
| 83 | \( 1 + (2.25 - 1.80i)T + (18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-4.43 - 12.6i)T + (-69.5 + 55.4i)T^{2} \) |
| 97 | \( 1 + (-15.9 - 1.79i)T + (94.5 + 21.5i)T^{2} \) |
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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
−13.08326254793527100105657611219, −12.41688481091539456229889419247, −11.21322426849591436630029801462, −10.38799519222940406702654923317, −9.165745727830918256070636000661, −8.556744907354015063729472761288, −7.18270118613233164222378829649, −5.23217976637814913624328489249, −3.83693173282142710796895718147, −1.83912522714894182089192413030,
1.74275820827702909713883713471, 4.50080748602990251216511208776, 6.05559276183720946347838429636, 7.28905530477442531431066883205, 7.84654211703921917748222185425, 9.299657814087729921108572489305, 10.44613419724467382624980935598, 11.12089459287741314999780693438, 12.80009356149206322170469984505, 13.87686365551472169077852009015
