L(s) = 1 | + (0.280 − 0.0752i)3-s + (2.21 − 0.332i)5-s + (1.13 − 2.39i)7-s + (−2.52 + 1.45i)9-s + (1.58 − 2.74i)11-s + (1.12 + 1.12i)13-s + (0.595 − 0.259i)15-s + (0.781 + 2.91i)17-s + (−1.03 − 1.79i)19-s + (0.137 − 0.756i)21-s + (2.21 + 0.593i)23-s + (4.77 − 1.47i)25-s + (−1.21 + 1.21i)27-s − 1.39i·29-s + (0.467 + 0.269i)31-s + ⋯ |
L(s) = 1 | + (0.162 − 0.0434i)3-s + (0.988 − 0.148i)5-s + (0.427 − 0.904i)7-s + (−0.841 + 0.485i)9-s + (0.477 − 0.826i)11-s + (0.312 + 0.312i)13-s + (0.153 − 0.0670i)15-s + (0.189 + 0.706i)17-s + (−0.237 − 0.411i)19-s + (0.0300 − 0.165i)21-s + (0.461 + 0.123i)23-s + (0.955 − 0.294i)25-s + (−0.233 + 0.233i)27-s − 0.259i·29-s + (0.0839 + 0.0484i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 + 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.922 + 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.53457 - 0.309108i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.53457 - 0.309108i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2.21 + 0.332i)T \) |
| 7 | \( 1 + (-1.13 + 2.39i)T \) |
good | 3 | \( 1 + (-0.280 + 0.0752i)T + (2.59 - 1.5i)T^{2} \) |
| 11 | \( 1 + (-1.58 + 2.74i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.12 - 1.12i)T + 13iT^{2} \) |
| 17 | \( 1 + (-0.781 - 2.91i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (1.03 + 1.79i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.21 - 0.593i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + 1.39iT - 29T^{2} \) |
| 31 | \( 1 + (-0.467 - 0.269i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.72 - 6.45i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + 1.82iT - 41T^{2} \) |
| 43 | \( 1 + (6.50 - 6.50i)T - 43iT^{2} \) |
| 47 | \( 1 + (12.4 + 3.32i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-2.89 - 10.8i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (-2.55 + 4.43i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (12.2 - 7.09i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.08 - 0.826i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 7.61T + 71T^{2} \) |
| 73 | \( 1 + (-3.08 + 0.827i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (14.7 - 8.52i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.62 - 4.62i)T + 83iT^{2} \) |
| 89 | \( 1 + (8.86 + 15.3i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.71 + 3.71i)T - 97iT^{2} \) |
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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
−11.56229155838927148259642048029, −10.88866741480905128565686324010, −9.976689263643922940811796504566, −8.823136931913687980056019338237, −8.150722985710667582074866095452, −6.74730215838553304352665529318, −5.82191601036865216994365448303, −4.65409387667470370467046673194, −3.15397104368441932288979933045, −1.50676413050410615482549187809,
1.92143101925075073526362094565, 3.17789515849469283662211865784, 4.97740265732237235804561936891, 5.86376137675178577117679302457, 6.84585521035114487350298185555, 8.297986156465125902623277311342, 9.135727391963863548925843126614, 9.833076424505313486898411965138, 11.03403701313964891888449822324, 11.94694908155798860009551307736