| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s + 2·11-s − 2·13-s − 14-s + 16-s + 8·17-s − 2·19-s + 2·22-s − 2·26-s − 28-s + 6·29-s + 6·31-s + 32-s + 8·34-s − 8·37-s − 2·38-s − 6·41-s − 8·43-s + 2·44-s + 4·47-s + 49-s − 2·52-s − 2·53-s − 56-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s + 0.603·11-s − 0.554·13-s − 0.267·14-s + 1/4·16-s + 1.94·17-s − 0.458·19-s + 0.426·22-s − 0.392·26-s − 0.188·28-s + 1.11·29-s + 1.07·31-s + 0.176·32-s + 1.37·34-s − 1.31·37-s − 0.324·38-s − 0.937·41-s − 1.21·43-s + 0.301·44-s + 0.583·47-s + 1/7·49-s − 0.277·52-s − 0.274·53-s − 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.052227211\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.052227211\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−8.475238782487588287930160177207, −7.966250676695425034602702674978, −6.84082538028308130788476792675, −6.56987499478625730715997159610, −5.45433411349147032995111530050, −4.96853931903917398920229743910, −3.84163331242361600013300679743, −3.27219204913396451138478631958, −2.23883333457531068124681597087, −0.985802799512964965931522056329,
0.985802799512964965931522056329, 2.23883333457531068124681597087, 3.27219204913396451138478631958, 3.84163331242361600013300679743, 4.96853931903917398920229743910, 5.45433411349147032995111530050, 6.56987499478625730715997159610, 6.84082538028308130788476792675, 7.966250676695425034602702674978, 8.475238782487588287930160177207
