| L(s) = 1 | − 2-s − 3-s + 4-s + 2·5-s + 6-s − 8-s + 9-s − 2·10-s − 11-s − 12-s − 2·15-s + 16-s + 17-s − 18-s + 6·19-s + 2·20-s + 22-s + 4·23-s + 24-s − 25-s − 27-s − 6·29-s + 2·30-s + 8·31-s − 32-s + 33-s − 34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.301·11-s − 0.288·12-s − 0.516·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 1.37·19-s + 0.447·20-s + 0.213·22-s + 0.834·23-s + 0.204·24-s − 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.365·30-s + 1.43·31-s − 0.176·32-s + 0.174·33-s − 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.477470826\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.477470826\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 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
−13.25042535801673, −12.55827479369371, −12.05047482025122, −11.68430168533635, −10.99942730429238, −10.89733421226277, −10.17633209042023, −9.735539414735428, −9.386285040205665, −9.172796577861158, −8.179696484052011, −7.891410493398157, −7.437866157664414, −6.766610017133585, −6.372020223379230, −5.808102107927226, −5.476776780079745, −4.855701791831837, −4.383455110643497, −3.409693969801514, −2.982018573566980, −2.383604059852546, −1.541061409957248, −1.262238251648070, −0.4201099038408691,
0.4201099038408691, 1.262238251648070, 1.541061409957248, 2.383604059852546, 2.982018573566980, 3.409693969801514, 4.383455110643497, 4.855701791831837, 5.476776780079745, 5.808102107927226, 6.372020223379230, 6.766610017133585, 7.437866157664414, 7.891410493398157, 8.179696484052011, 9.172796577861158, 9.386285040205665, 9.735539414735428, 10.17633209042023, 10.89733421226277, 10.99942730429238, 11.68430168533635, 12.05047482025122, 12.55827479369371, 13.25042535801673
