| L(s) = 1 | + 2-s + 4-s − 2·7-s + 8-s + 3·11-s + 13-s − 2·14-s + 16-s − 3·17-s + 8·19-s + 3·22-s + 3·23-s + 26-s − 2·28-s + 9·29-s − 7·31-s + 32-s − 3·34-s − 2·37-s + 8·38-s + 12·41-s + 7·43-s + 3·44-s + 3·46-s − 3·47-s − 3·49-s + 52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s + 0.904·11-s + 0.277·13-s − 0.534·14-s + 1/4·16-s − 0.727·17-s + 1.83·19-s + 0.639·22-s + 0.625·23-s + 0.196·26-s − 0.377·28-s + 1.67·29-s − 1.25·31-s + 0.176·32-s − 0.514·34-s − 0.328·37-s + 1.29·38-s + 1.87·41-s + 1.06·43-s + 0.452·44-s + 0.442·46-s − 0.437·47-s − 3/7·49-s + 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.648785563\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.648785563\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 18 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−9.429245595815721672181577127890, −9.072503855710526131661237974419, −7.77525488848171336433338902364, −6.96206917457341163684760702823, −6.29697405356291031337465683385, −5.44624806543657551278179807260, −4.42905986906434781866538981809, −3.52758168075608665434978537597, −2.69679316478050875122526015205, −1.15765950034440094850851077493,
1.15765950034440094850851077493, 2.69679316478050875122526015205, 3.52758168075608665434978537597, 4.42905986906434781866538981809, 5.44624806543657551278179807260, 6.29697405356291031337465683385, 6.96206917457341163684760702823, 7.77525488848171336433338902364, 9.072503855710526131661237974419, 9.429245595815721672181577127890
