L(s) = 1 | + 2·2-s − 3-s + 2·4-s + 5-s − 2·6-s + 9-s + 2·10-s + 5·11-s − 2·12-s + 13-s − 15-s − 4·16-s + 7·17-s + 2·18-s + 6·19-s + 2·20-s + 10·22-s + 3·23-s + 25-s + 2·26-s − 27-s + 2·29-s − 2·30-s − 2·31-s − 8·32-s − 5·33-s + 14·34-s + ⋯ |
L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s + 0.447·5-s − 0.816·6-s + 1/3·9-s + 0.632·10-s + 1.50·11-s − 0.577·12-s + 0.277·13-s − 0.258·15-s − 16-s + 1.69·17-s + 0.471·18-s + 1.37·19-s + 0.447·20-s + 2.13·22-s + 0.625·23-s + 1/5·25-s + 0.392·26-s − 0.192·27-s + 0.371·29-s − 0.365·30-s − 0.359·31-s − 1.41·32-s − 0.870·33-s + 2.40·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9555 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9555 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.104767090\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.104767090\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−7.33336734150810117899384110898, −6.63766955231328651971714453049, −6.22825685841877325575095618557, −5.42070052673462305634659242884, −5.12809484155897799374975575742, −4.27349695575836161639954637710, −3.42523128173611276512755811052, −3.12479027254654742175262146773, −1.71851321747435352467993608536, −0.976920050795348365188556174843,
0.976920050795348365188556174843, 1.71851321747435352467993608536, 3.12479027254654742175262146773, 3.42523128173611276512755811052, 4.27349695575836161639954637710, 5.12809484155897799374975575742, 5.42070052673462305634659242884, 6.22825685841877325575095618557, 6.63766955231328651971714453049, 7.33336734150810117899384110898