| L(s) = 1 | + 3-s + 2·7-s + 9-s + 6·11-s − 2·13-s − 2·19-s + 2·21-s + 6·23-s + 27-s + 6·29-s − 8·31-s + 6·33-s − 2·37-s − 2·39-s − 6·41-s + 43-s + 6·47-s − 3·49-s + 6·53-s − 2·57-s − 6·59-s + 8·61-s + 2·63-s − 4·67-s + 6·69-s + 4·73-s + 12·77-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.755·7-s + 1/3·9-s + 1.80·11-s − 0.554·13-s − 0.458·19-s + 0.436·21-s + 1.25·23-s + 0.192·27-s + 1.11·29-s − 1.43·31-s + 1.04·33-s − 0.328·37-s − 0.320·39-s − 0.937·41-s + 0.152·43-s + 0.875·47-s − 3/7·49-s + 0.824·53-s − 0.264·57-s − 0.781·59-s + 1.02·61-s + 0.251·63-s − 0.488·67-s + 0.722·69-s + 0.468·73-s + 1.36·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.221248652\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.221248652\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.51690247761411, −14.07217407473277, −13.66840669032398, −12.87388681144215, −12.48517915308561, −11.91543711729783, −11.41584430630516, −10.99120508145039, −10.28286062382500, −9.782585239889307, −9.126385949894792, −8.696105358600252, −8.487210422026291, −7.444390795817189, −7.244168718543780, −6.593582891470374, −6.040903019803831, −5.149659432131325, −4.723071168260720, −4.062247318707720, −3.546081324499032, −2.838300412882439, −1.997425822775843, −1.494495808759648, −0.7220951940626423,
0.7220951940626423, 1.494495808759648, 1.997425822775843, 2.838300412882439, 3.546081324499032, 4.062247318707720, 4.723071168260720, 5.149659432131325, 6.040903019803831, 6.593582891470374, 7.244168718543780, 7.444390795817189, 8.487210422026291, 8.696105358600252, 9.126385949894792, 9.782585239889307, 10.28286062382500, 10.99120508145039, 11.41584430630516, 11.91543711729783, 12.48517915308561, 12.87388681144215, 13.66840669032398, 14.07217407473277, 14.51690247761411
