L(s) = 1 | − 2-s + 4-s − 4·7-s − 8-s − 3·9-s + 4·11-s + 4·14-s + 16-s − 3·17-s + 3·18-s − 4·22-s + 4·23-s − 4·28-s − 29-s + 4·31-s − 32-s + 3·34-s − 3·36-s − 3·37-s − 9·41-s + 8·43-s + 4·44-s − 4·46-s + 8·47-s + 9·49-s + 9·53-s + 4·56-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.51·7-s − 0.353·8-s − 9-s + 1.20·11-s + 1.06·14-s + 1/4·16-s − 0.727·17-s + 0.707·18-s − 0.852·22-s + 0.834·23-s − 0.755·28-s − 0.185·29-s + 0.718·31-s − 0.176·32-s + 0.514·34-s − 1/2·36-s − 0.493·37-s − 1.40·41-s + 1.21·43-s + 0.603·44-s − 0.589·46-s + 1.16·47-s + 9/7·49-s + 1.23·53-s + 0.534·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 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
−7.14875484516170842119272073310, −6.97335867120383559248926848160, −6.09681643025387149079023286536, −5.77591656518001924720697938200, −4.55669324283618220387022236190, −3.63261086511153417657599996806, −3.03000710758906979378381787418, −2.26030095151491855898526338362, −1.00015221900735793101469407219, 0,
1.00015221900735793101469407219, 2.26030095151491855898526338362, 3.03000710758906979378381787418, 3.63261086511153417657599996806, 4.55669324283618220387022236190, 5.77591656518001924720697938200, 6.09681643025387149079023286536, 6.97335867120383559248926848160, 7.14875484516170842119272073310