L(s) = 1 | − 2-s + 3-s − 4-s − 2·5-s − 6-s + 4·7-s + 3·8-s + 9-s + 2·10-s − 11-s − 12-s − 2·13-s − 4·14-s − 2·15-s − 16-s + 2·17-s − 18-s + 2·20-s + 4·21-s + 22-s + 8·23-s + 3·24-s − 25-s + 2·26-s + 27-s − 4·28-s + 2·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.894·5-s − 0.408·6-s + 1.51·7-s + 1.06·8-s + 1/3·9-s + 0.632·10-s − 0.301·11-s − 0.288·12-s − 0.554·13-s − 1.06·14-s − 0.516·15-s − 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.447·20-s + 0.872·21-s + 0.213·22-s + 1.66·23-s + 0.612·24-s − 1/5·25-s + 0.392·26-s + 0.192·27-s − 0.755·28-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27753 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27753 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.665328749\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.665328749\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 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 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−15.17123881078544, −14.77221545839837, −14.25570145247551, −13.65365146797839, −13.31649032008915, −12.38147533307945, −12.08397655989371, −11.30589821271135, −10.90868657864677, −10.32372592048180, −9.733194685541141, −9.083052297673875, −8.563535909407787, −8.120092994386393, −7.683277369863402, −7.363619919244069, −6.553557563558221, −5.332613957041202, −4.894924852503408, −4.510859319031235, −3.747229005018218, −3.029239483912889, −2.105054256575642, −1.328984766309975, −0.6085456966138292,
0.6085456966138292, 1.328984766309975, 2.105054256575642, 3.029239483912889, 3.747229005018218, 4.510859319031235, 4.894924852503408, 5.332613957041202, 6.553557563558221, 7.363619919244069, 7.683277369863402, 8.120092994386393, 8.563535909407787, 9.083052297673875, 9.733194685541141, 10.32372592048180, 10.90868657864677, 11.30589821271135, 12.08397655989371, 12.38147533307945, 13.31649032008915, 13.65365146797839, 14.25570145247551, 14.77221545839837, 15.17123881078544