L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s + 11-s + 12-s − 6·13-s + 16-s + 5·17-s + 18-s + 6·19-s + 22-s + 23-s + 24-s − 5·25-s − 6·26-s + 27-s + 9·29-s − 6·31-s + 32-s + 33-s + 5·34-s + 36-s + 6·37-s + 6·38-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 + 0.301·11-s + 0.288·12-s − 1.66·13-s + 1/4·16-s + 1.21·17-s + 0.235·18-s + 1.37·19-s + 0.213·22-s + 0.208·23-s + 0.204·24-s − 25-s − 1.17·26-s + 0.192·27-s + 1.67·29-s − 1.07·31-s + 0.176·32-s + 0.174·33-s + 0.857·34-s + 1/6·36-s + 0.986·37-s + 0.973·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.270163485\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.270163485\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−7.65694925147766315429888627593, −7.41695611686945517688052630184, −6.66052328308385928849932942971, −5.62235681323745468638743858240, −5.17059702877625050156504531728, −4.34831853743474764106439445263, −3.52405513181818823316538020988, −2.86394011695279586144184033322, −2.09338309381785487030637769641, −0.948360650385425359388273894828,
0.948360650385425359388273894828, 2.09338309381785487030637769641, 2.86394011695279586144184033322, 3.52405513181818823316538020988, 4.34831853743474764106439445263, 5.17059702877625050156504531728, 5.62235681323745468638743858240, 6.66052328308385928849932942971, 7.41695611686945517688052630184, 7.65694925147766315429888627593