L(s) = 1 | + (−0.707 − 0.707i)2-s + (0.382 − 0.923i)3-s + 1.00i·4-s + (−0.382 + 0.923i)5-s + (−0.923 + 0.382i)6-s + (0.707 − 0.707i)7-s + (0.707 − 0.707i)8-s + (−0.707 − 0.707i)9-s + (0.923 − 0.382i)10-s + (0.923 + 0.382i)12-s + (1.30 + 1.30i)13-s − 1.00·14-s + (0.707 + 0.707i)15-s − 1.00·16-s + i·18-s − 0.765i·19-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)2-s + (0.382 − 0.923i)3-s + 1.00i·4-s + (−0.382 + 0.923i)5-s + (−0.923 + 0.382i)6-s + (0.707 − 0.707i)7-s + (0.707 − 0.707i)8-s + (−0.707 − 0.707i)9-s + (0.923 − 0.382i)10-s + (0.923 + 0.382i)12-s + (1.30 + 1.30i)13-s − 1.00·14-s + (0.707 + 0.707i)15-s − 1.00·16-s + i·18-s − 0.765i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8127140362\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8127140362\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + (-0.382 + 0.923i)T \) |
| 5 | \( 1 + (0.382 - 0.923i)T \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
good | 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (-1.30 - 1.30i)T + iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + 0.765iT - T^{2} \) |
| 23 | \( 1 + (-1 + i)T - iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + 0.765T + T^{2} \) |
| 61 | \( 1 + 1.84T + T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 - 1.41iT - T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 - 1.41iT - T^{2} \) |
| 83 | \( 1 + (-0.541 + 0.541i)T - iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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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
−10.47273894365546179990825243489, −9.191722695227310098641564305698, −8.563429567315471317763893097629, −7.74239952910382920872685612701, −6.97289958916690344808545960582, −6.45289456093010333038197851981, −4.42476091110298767738284224309, −3.50255551713880824964302553645, −2.43594433320214708286483971695, −1.26079521219839814510303856886,
1.50985470321534075997042451947, 3.29815835664180795271384971436, 4.57950626846285614279986335162, 5.39869069388467044589086831587, 5.94383173349934888519950519421, 7.75373326226670262663843495521, 8.099807878916217670766129894382, 8.910482831749082906613586521224, 9.350918827467259465867299141608, 10.51259310371125956500564851379