L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 4·7-s − 8-s + 9-s + 4·11-s − 12-s − 13-s − 4·14-s + 16-s + 4·17-s − 18-s + 7·19-s − 4·21-s − 4·22-s − 4·23-s + 24-s + 26-s − 27-s + 4·28-s + 5·29-s + 4·31-s − 32-s − 4·33-s − 4·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s + 1.20·11-s − 0.288·12-s − 0.277·13-s − 1.06·14-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 1.60·19-s − 0.872·21-s − 0.852·22-s − 0.834·23-s + 0.204·24-s + 0.196·26-s − 0.192·27-s + 0.755·28-s + 0.928·29-s + 0.718·31-s − 0.176·32-s − 0.696·33-s − 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.452750646\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.452750646\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−9.220850738586000722454295393692, −8.326841886536800227463298234142, −7.69961945616265855221880805582, −7.00499618871619652580166774171, −6.02909682551866994311032321382, −5.22257577373108965022588114871, −4.41881629162078511791323903590, −3.23851281546592840735313226204, −1.73249182974967552212476879741, −1.03973125861146578530883130890,
1.03973125861146578530883130890, 1.73249182974967552212476879741, 3.23851281546592840735313226204, 4.41881629162078511791323903590, 5.22257577373108965022588114871, 6.02909682551866994311032321382, 7.00499618871619652580166774171, 7.69961945616265855221880805582, 8.326841886536800227463298234142, 9.220850738586000722454295393692