| L(s) = 1 | − 2-s − 2·3-s + 4-s + 5-s + 2·6-s − 7-s − 8-s + 9-s − 10-s + 11-s − 2·12-s + 14-s − 2·15-s + 16-s − 18-s + 4·19-s + 20-s + 2·21-s − 22-s + 2·24-s + 25-s + 4·27-s − 28-s − 6·29-s + 2·30-s + 10·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.447·5-s + 0.816·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s − 0.577·12-s + 0.267·14-s − 0.516·15-s + 1/4·16-s − 0.235·18-s + 0.917·19-s + 0.223·20-s + 0.436·21-s − 0.213·22-s + 0.408·24-s + 1/5·25-s + 0.769·27-s − 0.188·28-s − 1.11·29-s + 0.365·30-s + 1.79·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 130130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.69701011632504, −13.07721977456486, −12.69054513233206, −12.07347164294850, −11.76158755441283, −11.16879321179151, −11.01051083111434, −10.22847964269221, −9.884645741731701, −9.540264996305769, −8.898724677362939, −8.447397658779734, −7.754525066497934, −7.284807445602831, −6.687116895577974, −6.268163513546369, −5.851630272191699, −5.373205356482936, −4.769293854500851, −4.178377542509381, −3.291250211106222, −2.855210752598949, −2.076125479765732, −1.294167826730127, −0.7819775724709623, 0,
0.7819775724709623, 1.294167826730127, 2.076125479765732, 2.855210752598949, 3.291250211106222, 4.178377542509381, 4.769293854500851, 5.373205356482936, 5.851630272191699, 6.268163513546369, 6.687116895577974, 7.284807445602831, 7.754525066497934, 8.447397658779734, 8.898724677362939, 9.540264996305769, 9.884645741731701, 10.22847964269221, 11.01051083111434, 11.16879321179151, 11.76158755441283, 12.07347164294850, 12.69054513233206, 13.07721977456486, 13.69701011632504
