| L(s) = 1 | − 2-s + 4-s + 5-s + 2·7-s − 8-s − 10-s − 13-s − 2·14-s + 16-s + 17-s − 19-s + 20-s + 6·23-s + 25-s + 26-s + 2·28-s + 3·29-s + 5·31-s − 32-s − 34-s + 2·35-s + 8·37-s + 38-s − 40-s − 6·41-s − 10·43-s − 6·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.755·7-s − 0.353·8-s − 0.316·10-s − 0.277·13-s − 0.534·14-s + 1/4·16-s + 0.242·17-s − 0.229·19-s + 0.223·20-s + 1.25·23-s + 1/5·25-s + 0.196·26-s + 0.377·28-s + 0.557·29-s + 0.898·31-s − 0.176·32-s − 0.171·34-s + 0.338·35-s + 1.31·37-s + 0.162·38-s − 0.158·40-s − 0.937·41-s − 1.52·43-s − 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.484842653\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.484842653\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−9.519646065076311179766613591631, −8.567757106912053419231683120503, −8.080941537947816599267611130841, −7.09404536827976955997053532459, −6.40144355826335323431913115367, −5.33319452159623685218826002122, −4.58632421025304540433634126608, −3.16800465585064564790696086653, −2.13651444680928063495502053275, −1.00838220071117878119878324340,
1.00838220071117878119878324340, 2.13651444680928063495502053275, 3.16800465585064564790696086653, 4.58632421025304540433634126608, 5.33319452159623685218826002122, 6.40144355826335323431913115367, 7.09404536827976955997053532459, 8.080941537947816599267611130841, 8.567757106912053419231683120503, 9.519646065076311179766613591631
