L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 2·7-s − 8-s + 9-s + 10-s + 12-s − 13-s + 2·14-s − 15-s + 16-s − 17-s − 18-s − 4·19-s − 20-s − 2·21-s − 9·23-s − 24-s − 4·25-s + 26-s + 27-s − 2·28-s − 6·29-s + 30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s − 0.277·13-s + 0.534·14-s − 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.917·19-s − 0.223·20-s − 0.436·21-s − 1.87·23-s − 0.204·24-s − 4/5·25-s + 0.196·26-s + 0.192·27-s − 0.377·28-s − 1.11·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160446 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160446 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 13 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 15 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−13.27610350204723, −12.99818159463945, −12.73306189314476, −11.99085001661631, −11.46416121876549, −11.28570648279372, −10.45245850973054, −10.00320060730405, −9.741610919299506, −9.232149741804878, −8.662155188826946, −8.230353856176489, −7.787021458974921, −7.344795518007869, −6.773188977751695, −6.318696176682059, −5.736242165218714, −5.215661947272290, −4.184523159874898, −3.914645029080987, −3.495462877231697, −2.640583993364472, −2.098822791177851, −1.753459987456170, −0.5763572115865194, 0,
0.5763572115865194, 1.753459987456170, 2.098822791177851, 2.640583993364472, 3.495462877231697, 3.914645029080987, 4.184523159874898, 5.215661947272290, 5.736242165218714, 6.318696176682059, 6.773188977751695, 7.344795518007869, 7.787021458974921, 8.230353856176489, 8.662155188826946, 9.232149741804878, 9.741610919299506, 10.00320060730405, 10.45245850973054, 11.28570648279372, 11.46416121876549, 11.99085001661631, 12.73306189314476, 12.99818159463945, 13.27610350204723