| L(s) = 1 | − 2-s − 3-s + 4-s − 2·5-s + 6-s + 7-s − 8-s + 9-s + 2·10-s + 11-s − 12-s − 13-s − 14-s + 2·15-s + 16-s + 2·17-s − 18-s + 4·19-s − 2·20-s − 21-s − 22-s − 4·23-s + 24-s − 25-s + 26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.301·11-s − 0.288·12-s − 0.277·13-s − 0.267·14-s + 0.516·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.917·19-s − 0.447·20-s − 0.218·21-s − 0.213·22-s − 0.834·23-s + 0.204·24-s − 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.60026975176281297804420938805, −7.34511998891413123319227402039, −6.39822098227337342623672273563, −5.64543952698411022628636519983, −4.90373837746016070464579786047, −3.98674282890364879068493380075, −3.30319751128433136721620527175, −2.07680484356349921343674701987, −1.08221155862759820490397465421, 0,
1.08221155862759820490397465421, 2.07680484356349921343674701987, 3.30319751128433136721620527175, 3.98674282890364879068493380075, 4.90373837746016070464579786047, 5.64543952698411022628636519983, 6.39822098227337342623672273563, 7.34511998891413123319227402039, 7.60026975176281297804420938805
