L(s) = 1 | + (−0.707 − 0.707i)2-s − i·3-s + 1.00i·4-s + (2.16 − 2.16i)5-s + (−0.707 + 0.707i)6-s + (−0.292 − 2.62i)7-s + (0.707 − 0.707i)8-s − 9-s − 3.06·10-s + (0.516 − 0.516i)11-s + 1.00·12-s + (−3.60 − 0.0469i)13-s + (−1.65 + 2.06i)14-s + (−2.16 − 2.16i)15-s − 1.00·16-s + 2.57·17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s − 0.577i·3-s + 0.500i·4-s + (0.969 − 0.969i)5-s + (−0.288 + 0.288i)6-s + (−0.110 − 0.993i)7-s + (0.250 − 0.250i)8-s − 0.333·9-s − 0.969·10-s + (0.155 − 0.155i)11-s + 0.288·12-s + (−0.999 − 0.0130i)13-s + (−0.441 + 0.552i)14-s + (−0.559 − 0.559i)15-s − 0.250·16-s + 0.624·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.831 + 0.555i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.831 + 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.341901 - 1.12659i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.341901 - 1.12659i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 + (0.292 + 2.62i)T \) |
| 13 | \( 1 + (3.60 + 0.0469i)T \) |
good | 5 | \( 1 + (-2.16 + 2.16i)T - 5iT^{2} \) |
| 11 | \( 1 + (-0.516 + 0.516i)T - 11iT^{2} \) |
| 17 | \( 1 - 2.57T + 17T^{2} \) |
| 19 | \( 1 + (-3.82 + 3.82i)T - 19iT^{2} \) |
| 23 | \( 1 - 6.97iT - 23T^{2} \) |
| 29 | \( 1 + 6.32T + 29T^{2} \) |
| 31 | \( 1 + (-4.33 + 4.33i)T - 31iT^{2} \) |
| 37 | \( 1 + (3.58 - 3.58i)T - 37iT^{2} \) |
| 41 | \( 1 + (-0.176 + 0.176i)T - 41iT^{2} \) |
| 43 | \( 1 + 7.89iT - 43T^{2} \) |
| 47 | \( 1 + (1.41 + 1.41i)T + 47iT^{2} \) |
| 53 | \( 1 + 0.514T + 53T^{2} \) |
| 59 | \( 1 + (-3.17 - 3.17i)T + 59iT^{2} \) |
| 61 | \( 1 - 6.41iT - 61T^{2} \) |
| 67 | \( 1 + (3.96 + 3.96i)T + 67iT^{2} \) |
| 71 | \( 1 + (-8.12 - 8.12i)T + 71iT^{2} \) |
| 73 | \( 1 + (11.6 + 11.6i)T + 73iT^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 + (1.01 - 1.01i)T - 83iT^{2} \) |
| 89 | \( 1 + (1.10 + 1.10i)T + 89iT^{2} \) |
| 97 | \( 1 + (-9.34 + 9.34i)T - 97iT^{2} \) |
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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
−10.18497368900383831200220019414, −9.618348532502788379958008778479, −8.908327789123190996661483175081, −7.68477917174843258547490349646, −7.14171811564091293997731666907, −5.75977785496276644311851110761, −4.83452391987979430777098958944, −3.37717323093755954360744814526, −1.91038143177369594306637872741, −0.817085786820111482479578166744,
2.10536962244088200212049556287, 3.15304879151353071532017154673, 4.92162255691571928501886047458, 5.78205271537268502703952325170, 6.50842377609731636161582344667, 7.56728735840662245394574499898, 8.643814060367003140781184372701, 9.669981950210086756435154334004, 9.927398262562887359225559069501, 10.82665236399669015239707459594