| L(s) = 1 | − 2-s + 4-s − 5-s + 2·7-s − 8-s + 10-s − 11-s − 13-s − 2·14-s + 16-s + 19-s − 20-s + 22-s + 25-s + 26-s + 2·28-s − 8·29-s − 5·31-s − 32-s − 2·35-s + 3·37-s − 38-s + 40-s + 4·41-s − 9·43-s − 44-s − 6·47-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 − 0.301·11-s − 0.277·13-s − 0.534·14-s + 1/4·16-s + 0.229·19-s − 0.223·20-s + 0.213·22-s + 1/5·25-s + 0.196·26-s + 0.377·28-s − 1.48·29-s − 0.898·31-s − 0.176·32-s − 0.338·35-s + 0.493·37-s − 0.162·38-s + 0.158·40-s + 0.624·41-s − 1.37·43-s − 0.150·44-s − 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6767039441\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6767039441\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 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
−12.60519968558644, −11.85680658028199, −11.54596474377959, −11.30530596505546, −10.79037793514249, −10.29965570264753, −9.825579519215666, −9.388466333048565, −8.884520828157120, −8.418069396694778, −7.978133087025681, −7.579575362241645, −7.211088319735435, −6.703179321505957, −6.079698191251764, −5.471363821795323, −5.147497697408355, −4.533601794105322, −3.949465386673287, −3.410408497050896, −2.864266890224746, −2.129895124022902, −1.730904112096878, −1.089775507442350, −0.2551615891946762,
0.2551615891946762, 1.089775507442350, 1.730904112096878, 2.129895124022902, 2.864266890224746, 3.410408497050896, 3.949465386673287, 4.533601794105322, 5.147497697408355, 5.471363821795323, 6.079698191251764, 6.703179321505957, 7.211088319735435, 7.579575362241645, 7.978133087025681, 8.418069396694778, 8.884520828157120, 9.388466333048565, 9.825579519215666, 10.29965570264753, 10.79037793514249, 11.30530596505546, 11.54596474377959, 11.85680658028199, 12.60519968558644
