| L(s) = 1 | + (−1.22 + 1.22i)3-s + (−3.67 − 3.67i)7-s − 2.99i·9-s + (1.22 − 1.22i)13-s − 7i·19-s + 9·21-s + (3.67 + 3.67i)27-s − 11·31-s + (−4.89 − 4.89i)37-s + 2.99i·39-s + (1.22 − 1.22i)43-s + 20i·49-s + (8.57 + 8.57i)57-s − 61-s + (−11.0 + 11.0i)63-s + ⋯ |
| L(s) = 1 | + (−0.707 + 0.707i)3-s + (−1.38 − 1.38i)7-s − 0.999i·9-s + (0.339 − 0.339i)13-s − 1.60i·19-s + 1.96·21-s + (0.707 + 0.707i)27-s − 1.97·31-s + (−0.805 − 0.805i)37-s + 0.480i·39-s + (0.186 − 0.186i)43-s + 2.85i·49-s + (1.13 + 1.13i)57-s − 0.128·61-s + (−1.38 + 1.38i)63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.130 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.130 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.368743 - 0.420575i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.368743 - 0.420575i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.22 - 1.22i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (3.67 + 3.67i)T + 7iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + (-1.22 + 1.22i)T - 13iT^{2} \) |
| 17 | \( 1 - 17iT^{2} \) |
| 19 | \( 1 + 7iT - 19T^{2} \) |
| 23 | \( 1 + 23iT^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 11T + 31T^{2} \) |
| 37 | \( 1 + (4.89 + 4.89i)T + 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + (-1.22 + 1.22i)T - 43iT^{2} \) |
| 47 | \( 1 - 47iT^{2} \) |
| 53 | \( 1 + 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 + (-8.57 - 8.57i)T + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (-9.79 + 9.79i)T - 73iT^{2} \) |
| 79 | \( 1 - 4iT - 79T^{2} \) |
| 83 | \( 1 + 83iT^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + (3.67 + 3.67i)T + 97iT^{2} \) |
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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
−11.11860238171895224302109532218, −10.66620269302834580076559266176, −9.724336581674903305675141520519, −9.014468345573923964701132788590, −7.27063806508378292236530837756, −6.62190355994087854172886235568, −5.45229441999026992233723255918, −4.15313930557551341846262812536, −3.29679380227759339369699130625, −0.43487692495510207052947178929,
2.00105029732420890941231738876, 3.49490087929753217566214985753, 5.38283383068591988890539478739, 6.06431871183911066777028233098, 6.88498259536766895505327696159, 8.166704774242188692017644598861, 9.188105604863573297427055625240, 10.13204404431441702005750004303, 11.25691716045323270066672446965, 12.27220848654636882662166615928
