L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 7-s − 8-s + 9-s + 2·11-s − 12-s − 13-s − 14-s + 16-s + 17-s − 18-s + 4·19-s − 21-s − 2·22-s − 7·23-s + 24-s + 26-s − 27-s + 28-s + 29-s + 3·31-s − 32-s − 2·33-s − 34-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.603·11-s − 0.288·12-s − 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.917·19-s − 0.218·21-s − 0.426·22-s − 1.45·23-s + 0.204·24-s + 0.196·26-s − 0.192·27-s + 0.188·28-s + 0.185·29-s + 0.538·31-s − 0.176·32-s − 0.348·33-s − 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.002711243\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.002711243\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−9.867884616148654764319249926263, −9.260468983053516807245881636264, −8.158214749417890256597687504313, −7.55252193275581649651700207024, −6.56557884551623356952409312838, −5.80952482446894196152644579304, −4.77770790573633599440232036463, −3.65078540287386924344518852742, −2.19259007680211571616293034668, −0.907613785551433006810129340306,
0.907613785551433006810129340306, 2.19259007680211571616293034668, 3.65078540287386924344518852742, 4.77770790573633599440232036463, 5.80952482446894196152644579304, 6.56557884551623356952409312838, 7.55252193275581649651700207024, 8.158214749417890256597687504313, 9.260468983053516807245881636264, 9.867884616148654764319249926263