L(s) = 1 | + 2-s − 3-s − 4-s − 6-s − 3·8-s + 9-s − 11-s + 12-s + 4·13-s − 16-s + 3·17-s + 18-s + 19-s − 22-s − 23-s + 3·24-s − 5·25-s + 4·26-s − 27-s − 5·29-s − 10·31-s + 5·32-s + 33-s + 3·34-s − 36-s − 11·37-s + 38-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s + 1.10·13-s − 1/4·16-s + 0.727·17-s + 0.235·18-s + 0.229·19-s − 0.213·22-s − 0.208·23-s + 0.612·24-s − 25-s + 0.784·26-s − 0.192·27-s − 0.928·29-s − 1.79·31-s + 0.883·32-s + 0.174·33-s + 0.514·34-s − 1/6·36-s − 1.80·37-s + 0.162·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−9.114296082745761695321526454099, −8.169348363883784297869275153314, −7.34700480399560877109022917480, −6.17908347755492977195574256173, −5.66373182932450284848741323339, −4.96119631319196246301895597812, −3.88057852905744129610682622943, −3.32924483674236257523401238797, −1.64311325390777105977223172816, 0,
1.64311325390777105977223172816, 3.32924483674236257523401238797, 3.88057852905744129610682622943, 4.96119631319196246301895597812, 5.66373182932450284848741323339, 6.17908347755492977195574256173, 7.34700480399560877109022917480, 8.169348363883784297869275153314, 9.114296082745761695321526454099