L(s) = 1 | − 2·3-s + 4·7-s + 9-s − 2·11-s − 2·13-s + 2·17-s + 2·19-s − 8·21-s − 4·23-s + 4·27-s − 6·29-s + 4·33-s − 10·37-s + 4·39-s − 6·41-s − 6·43-s + 8·47-s + 9·49-s − 4·51-s + 6·53-s − 4·57-s + 14·59-s + 2·61-s + 4·63-s − 10·67-s + 8·69-s + 12·71-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 1.51·7-s + 1/3·9-s − 0.603·11-s − 0.554·13-s + 0.485·17-s + 0.458·19-s − 1.74·21-s − 0.834·23-s + 0.769·27-s − 1.11·29-s + 0.696·33-s − 1.64·37-s + 0.640·39-s − 0.937·41-s − 0.914·43-s + 1.16·47-s + 9/7·49-s − 0.560·51-s + 0.824·53-s − 0.529·57-s + 1.82·59-s + 0.256·61-s + 0.503·63-s − 1.22·67-s + 0.963·69-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−8.212002059885944684861021878477, −7.50419758061409482324369399006, −6.84978263741944475168127235714, −5.63664495018734253206101527190, −5.36997692852317792358039338755, −4.72524620355206898427648612891, −3.69802392152085758411183661848, −2.35167390023705158699342586043, −1.36685449629542271877328763705, 0,
1.36685449629542271877328763705, 2.35167390023705158699342586043, 3.69802392152085758411183661848, 4.72524620355206898427648612891, 5.36997692852317792358039338755, 5.63664495018734253206101527190, 6.84978263741944475168127235714, 7.50419758061409482324369399006, 8.212002059885944684861021878477