| L(s) = 1 | + (−1.31 − 0.525i)2-s + (−0.564 − 0.898i)3-s + (1.44 + 1.37i)4-s + (1.84 + 1.47i)5-s + (0.269 + 1.47i)6-s + (0.762 + 0.174i)7-s + (−1.17 − 2.57i)8-s + (0.812 − 1.68i)9-s + (−1.65 − 2.90i)10-s + (1.41 − 4.03i)11-s + (0.422 − 2.08i)12-s + (1.14 + 2.38i)13-s + (−0.909 − 0.629i)14-s + (0.281 − 2.49i)15-s + (0.193 + 3.99i)16-s + (−0.864 − 0.864i)17-s + ⋯ |
| L(s) = 1 | + (−0.928 − 0.371i)2-s + (−0.326 − 0.518i)3-s + (0.723 + 0.689i)4-s + (0.827 + 0.659i)5-s + (0.109 + 0.602i)6-s + (0.288 + 0.0657i)7-s + (−0.415 − 0.909i)8-s + (0.270 − 0.562i)9-s + (−0.522 − 0.919i)10-s + (0.425 − 1.21i)11-s + (0.121 − 0.600i)12-s + (0.318 + 0.661i)13-s + (−0.243 − 0.168i)14-s + (0.0725 − 0.644i)15-s + (0.0482 + 0.998i)16-s + (−0.209 − 0.209i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.777 + 0.629i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.777 + 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.723560 - 0.256217i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.723560 - 0.256217i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.31 + 0.525i)T \) |
| 29 | \( 1 + (-1.27 + 5.23i)T \) |
| good | 3 | \( 1 + (0.564 + 0.898i)T + (-1.30 + 2.70i)T^{2} \) |
| 5 | \( 1 + (-1.84 - 1.47i)T + (1.11 + 4.87i)T^{2} \) |
| 7 | \( 1 + (-0.762 - 0.174i)T + (6.30 + 3.03i)T^{2} \) |
| 11 | \( 1 + (-1.41 + 4.03i)T + (-8.60 - 6.85i)T^{2} \) |
| 13 | \( 1 + (-1.14 - 2.38i)T + (-8.10 + 10.1i)T^{2} \) |
| 17 | \( 1 + (0.864 + 0.864i)T + 17iT^{2} \) |
| 19 | \( 1 + (-3.53 - 2.22i)T + (8.24 + 17.1i)T^{2} \) |
| 23 | \( 1 + (5.08 - 4.05i)T + (5.11 - 22.4i)T^{2} \) |
| 31 | \( 1 + (-0.840 - 7.45i)T + (-30.2 + 6.89i)T^{2} \) |
| 37 | \( 1 + (3.11 - 1.08i)T + (28.9 - 23.0i)T^{2} \) |
| 41 | \( 1 + (6.22 - 6.22i)T - 41iT^{2} \) |
| 43 | \( 1 + (-2.71 - 0.305i)T + (41.9 + 9.56i)T^{2} \) |
| 47 | \( 1 + (5.74 + 2.00i)T + (36.7 + 29.3i)T^{2} \) |
| 53 | \( 1 + (1.82 - 2.28i)T + (-11.7 - 51.6i)T^{2} \) |
| 59 | \( 1 - 3.52iT - 59T^{2} \) |
| 61 | \( 1 + (-6.36 + 4.00i)T + (26.4 - 54.9i)T^{2} \) |
| 67 | \( 1 + (10.8 + 5.23i)T + (41.7 + 52.3i)T^{2} \) |
| 71 | \( 1 + (-7.65 + 3.68i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (2.76 + 0.311i)T + (71.1 + 16.2i)T^{2} \) |
| 79 | \( 1 + (10.5 - 3.68i)T + (61.7 - 49.2i)T^{2} \) |
| 83 | \( 1 + (-17.0 + 3.88i)T + (74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (13.9 - 1.57i)T + (86.7 - 19.8i)T^{2} \) |
| 97 | \( 1 + (4.13 + 2.59i)T + (42.0 + 87.3i)T^{2} \) |
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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
−13.43627385057869789585458674828, −11.95430245292963544585521912195, −11.45890688121776979776319736896, −10.21235014033263607333487934809, −9.336186206774436648589959238359, −8.113403292906792341925240508684, −6.74219069992245625252028907922, −6.05185063851611801502403061959, −3.41563941075259642448706244430, −1.58147871880504336218845780593,
1.83539382469798929590472035525, 4.73792522778683200373541937178, 5.74648822136113371977263396923, 7.18947383638580103142568894223, 8.413684726960027122318365154425, 9.605092002614310752459638332729, 10.17148726368328628179678492330, 11.25662821688129471245693472769, 12.55372177270608825727535377774, 13.78313924230761384911719814496
