| L(s) = 1 | − 3·3-s + 4·7-s + 9·9-s − 72·11-s + 6·13-s − 38·17-s − 52·19-s − 12·21-s + 152·23-s − 27·27-s − 78·29-s − 120·31-s + 216·33-s + 150·37-s − 18·39-s + 362·41-s − 484·43-s + 280·47-s − 327·49-s + 114·51-s + 670·53-s + 156·57-s − 696·59-s + 222·61-s + 36·63-s − 4·67-s − 456·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.215·7-s + 1/3·9-s − 1.97·11-s + 0.128·13-s − 0.542·17-s − 0.627·19-s − 0.124·21-s + 1.37·23-s − 0.192·27-s − 0.499·29-s − 0.695·31-s + 1.13·33-s + 0.666·37-s − 0.0739·39-s + 1.37·41-s − 1.71·43-s + 0.868·47-s − 0.953·49-s + 0.313·51-s + 1.73·53-s + 0.362·57-s − 1.53·59-s + 0.465·61-s + 0.0719·63-s − 0.00729·67-s − 0.795·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.074026174\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.074026174\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p^{3} T^{2} \) |
| 11 | \( 1 + 72 T + p^{3} T^{2} \) |
| 13 | \( 1 - 6 T + p^{3} T^{2} \) |
| 17 | \( 1 + 38 T + p^{3} T^{2} \) |
| 19 | \( 1 + 52 T + p^{3} T^{2} \) |
| 23 | \( 1 - 152 T + p^{3} T^{2} \) |
| 29 | \( 1 + 78 T + p^{3} T^{2} \) |
| 31 | \( 1 + 120 T + p^{3} T^{2} \) |
| 37 | \( 1 - 150 T + p^{3} T^{2} \) |
| 41 | \( 1 - 362 T + p^{3} T^{2} \) |
| 43 | \( 1 + 484 T + p^{3} T^{2} \) |
| 47 | \( 1 - 280 T + p^{3} T^{2} \) |
| 53 | \( 1 - 670 T + p^{3} T^{2} \) |
| 59 | \( 1 + 696 T + p^{3} T^{2} \) |
| 61 | \( 1 - 222 T + p^{3} T^{2} \) |
| 67 | \( 1 + 4 T + p^{3} T^{2} \) |
| 71 | \( 1 + 96 T + p^{3} T^{2} \) |
| 73 | \( 1 + 178 T + p^{3} T^{2} \) |
| 79 | \( 1 - 8 p T + p^{3} T^{2} \) |
| 83 | \( 1 + 612 T + p^{3} T^{2} \) |
| 89 | \( 1 - 994 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1634 T + p^{3} T^{2} \) |
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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
−9.413546668410605712924156276610, −8.497310512738676789586005233622, −7.67479124765257075715448512391, −6.94819411649591222504256383414, −5.86472365838579624169533227947, −5.16711049966913734828213781660, −4.40244113680151271480742637498, −3.04375495940361895483561746749, −2.03237067325083737730616570480, −0.52752301244668289932876258540,
0.52752301244668289932876258540, 2.03237067325083737730616570480, 3.04375495940361895483561746749, 4.40244113680151271480742637498, 5.16711049966913734828213781660, 5.86472365838579624169533227947, 6.94819411649591222504256383414, 7.67479124765257075715448512391, 8.497310512738676789586005233622, 9.413546668410605712924156276610
