| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s − 4·11-s + 12-s + 16-s + 6·17-s + 18-s − 4·19-s − 4·22-s − 8·23-s + 24-s + 27-s + 6·29-s + 8·31-s + 32-s − 4·33-s + 6·34-s + 36-s − 10·37-s − 4·38-s + 6·41-s − 4·43-s − 4·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 1.20·11-s + 0.288·12-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.917·19-s − 0.852·22-s − 1.66·23-s + 0.204·24-s + 0.192·27-s + 1.11·29-s + 1.43·31-s + 0.176·32-s − 0.696·33-s + 1.02·34-s + 1/6·36-s − 1.64·37-s − 0.648·38-s + 0.937·41-s − 0.609·43-s − 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 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 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−15.60266757844918, −15.04199786416887, −14.40975621973169, −14.03807446263427, −13.56553255098709, −13.02058904086203, −12.37231074155717, −12.08626835931234, −11.47349245587686, −10.50625613902698, −10.16469914353556, −9.982079312335805, −8.842375612299158, −8.283725617222704, −7.924675752999207, −7.298171257049364, −6.616914557782054, −5.845637967284277, −5.511647833112978, −4.518833868031745, −4.278596504232596, −3.214528649235723, −2.905543577418157, −2.102842924927731, −1.321791979833916, 0,
1.321791979833916, 2.102842924927731, 2.905543577418157, 3.214528649235723, 4.278596504232596, 4.518833868031745, 5.511647833112978, 5.845637967284277, 6.616914557782054, 7.298171257049364, 7.924675752999207, 8.283725617222704, 8.842375612299158, 9.982079312335805, 10.16469914353556, 10.50625613902698, 11.47349245587686, 12.08626835931234, 12.37231074155717, 13.02058904086203, 13.56553255098709, 14.03807446263427, 14.40975621973169, 15.04199786416887, 15.60266757844918
