L(s) = 1 | + (−0.587 + 0.809i)2-s + (−3.01 + 0.979i)3-s + (−0.309 − 0.951i)4-s + (2.13 + 0.672i)5-s + (0.979 − 3.01i)6-s − i·7-s + (0.951 + 0.309i)8-s + (5.70 − 4.14i)9-s + (−1.79 + 1.33i)10-s + (4.60 + 3.34i)11-s + (1.86 + 2.56i)12-s + (−2.50 − 3.44i)13-s + (0.809 + 0.587i)14-s + (−7.08 + 0.0631i)15-s + (−0.809 + 0.587i)16-s + (2.28 + 0.740i)17-s + ⋯ |
L(s) = 1 | + (−0.415 + 0.572i)2-s + (−1.74 + 0.565i)3-s + (−0.154 − 0.475i)4-s + (0.953 + 0.300i)5-s + (0.399 − 1.23i)6-s − 0.377i·7-s + (0.336 + 0.109i)8-s + (1.90 − 1.38i)9-s + (−0.568 + 0.420i)10-s + (1.38 + 1.00i)11-s + (0.537 + 0.740i)12-s + (−0.694 − 0.955i)13-s + (0.216 + 0.157i)14-s + (−1.83 + 0.0163i)15-s + (−0.202 + 0.146i)16-s + (0.553 + 0.179i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.204 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.204 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.453191 + 0.557857i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.453191 + 0.557857i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.587 - 0.809i)T \) |
| 5 | \( 1 + (-2.13 - 0.672i)T \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 + (3.01 - 0.979i)T + (2.42 - 1.76i)T^{2} \) |
| 11 | \( 1 + (-4.60 - 3.34i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (2.50 + 3.44i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-2.28 - 0.740i)T + (13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (1.63 - 5.03i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (2.84 - 3.91i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (0.328 + 1.01i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-1.02 + 3.16i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-6.02 - 8.28i)T + (-11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (5.30 - 3.85i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 3.49iT - 43T^{2} \) |
| 47 | \( 1 + (-6.98 + 2.27i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-7.36 + 2.39i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (6.24 - 4.53i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-2.54 - 1.84i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-7.18 - 2.33i)T + (54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (3.97 + 12.2i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (2.55 - 3.51i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (1.75 + 5.39i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-8.31 - 2.70i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-9.09 - 6.61i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (1.33 - 0.434i)T + (78.4 - 57.0i)T^{2} \) |
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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
−11.69375366874940127051135092801, −10.47588058136331347389467328700, −9.983488390390988033973802517367, −9.507639264856641936943105151261, −7.69064370785809093680568851107, −6.63353167635856974621882761428, −6.02241349516355825827679953269, −5.16748061980794150514138643201, −4.06281577489695786220404541159, −1.36240737529507050605553947505,
0.828977024751787825615888453509, 2.13632996996568358881143223277, 4.36523803028135364828945857302, 5.47401550526431157786387620426, 6.36465667580191100223584424990, 7.06515960588602713788907064595, 8.737095372077286160098795004920, 9.502304527897529937708878727212, 10.55144738977695185623147476430, 11.31026595694557738329125240159