L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 10-s + 6·13-s + 16-s − 2·17-s + 20-s + 23-s + 25-s − 6·26-s − 6·29-s + 8·31-s − 32-s + 2·34-s + 10·37-s − 40-s + 6·41-s − 8·43-s − 46-s − 8·47-s − 7·49-s − 50-s + 6·52-s + 6·53-s + 6·58-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s + 1.66·13-s + 1/4·16-s − 0.485·17-s + 0.223·20-s + 0.208·23-s + 1/5·25-s − 1.17·26-s − 1.11·29-s + 1.43·31-s − 0.176·32-s + 0.342·34-s + 1.64·37-s − 0.158·40-s + 0.937·41-s − 1.21·43-s − 0.147·46-s − 1.16·47-s − 49-s − 0.141·50-s + 0.832·52-s + 0.824·53-s + 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.485324539\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.485324539\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−9.163748715075083439883265625087, −8.353436541667944420314126180276, −7.82607747383224319732934075551, −6.59173420845419525321982044776, −6.27390201097746932960763910477, −5.27353152311567031698307896518, −4.12490374907506956820694719254, −3.12238319622843730394498717450, −2.00107551151695888688219506151, −0.935045408651431349885382411049,
0.935045408651431349885382411049, 2.00107551151695888688219506151, 3.12238319622843730394498717450, 4.12490374907506956820694719254, 5.27353152311567031698307896518, 6.27390201097746932960763910477, 6.59173420845419525321982044776, 7.82607747383224319732934075551, 8.353436541667944420314126180276, 9.163748715075083439883265625087