| L(s) = 1 | + 2-s − 4-s − 5-s − 3·8-s − 10-s + 4·11-s − 2·13-s − 16-s − 2·17-s + 4·19-s + 20-s + 4·22-s + 25-s − 2·26-s + 2·29-s + 5·32-s − 2·34-s − 10·37-s + 4·38-s + 3·40-s − 10·41-s + 4·43-s − 4·44-s − 8·47-s − 7·49-s + 50-s + 2·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.447·5-s − 1.06·8-s − 0.316·10-s + 1.20·11-s − 0.554·13-s − 1/4·16-s − 0.485·17-s + 0.917·19-s + 0.223·20-s + 0.852·22-s + 1/5·25-s − 0.392·26-s + 0.371·29-s + 0.883·32-s − 0.342·34-s − 1.64·37-s + 0.648·38-s + 0.474·40-s − 1.56·41-s + 0.609·43-s − 0.603·44-s − 1.16·47-s − 49-s + 0.141·50-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9215908766\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9215908766\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.60334701825816031360567721288, −14.55874952579683322699316330788, −13.71457429111566649845275587943, −12.40658486355792546821392015265, −11.57318950916085697381955699128, −9.738851148469479131038939478590, −8.554588868470699894187777077269, −6.76955885158912340897756950945, −5.04860090093668311608273372623, −3.63412818236731287070311550143,
3.63412818236731287070311550143, 5.04860090093668311608273372623, 6.76955885158912340897756950945, 8.554588868470699894187777077269, 9.738851148469479131038939478590, 11.57318950916085697381955699128, 12.40658486355792546821392015265, 13.71457429111566649845275587943, 14.55874952579683322699316330788, 15.60334701825816031360567721288
