L(s) = 1 | − 3-s − 7-s + 9-s + 2·11-s − 2·17-s + 21-s + 23-s − 27-s + 2·29-s + 6·31-s − 2·33-s − 4·37-s + 6·41-s − 4·43-s − 12·47-s + 49-s + 2·51-s + 6·59-s − 2·61-s − 63-s + 8·67-s − 69-s + 8·71-s − 2·77-s + 10·79-s + 81-s − 4·83-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.603·11-s − 0.485·17-s + 0.218·21-s + 0.208·23-s − 0.192·27-s + 0.371·29-s + 1.07·31-s − 0.348·33-s − 0.657·37-s + 0.937·41-s − 0.609·43-s − 1.75·47-s + 1/7·49-s + 0.280·51-s + 0.781·59-s − 0.256·61-s − 0.125·63-s + 0.977·67-s − 0.120·69-s + 0.949·71-s − 0.227·77-s + 1.12·79-s + 1/9·81-s − 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−13.13468886593905, −13.00553316883597, −12.22543511369940, −11.96599423499790, −11.52267206998452, −10.95577369552714, −10.61793474518439, −9.979621518450284, −9.629883511980523, −9.159638914106596, −8.572456840627392, −8.097040886271201, −7.584408446460277, −6.841669678766806, −6.457457790887647, −6.366199601357197, −5.411803557713758, −5.112014812099292, −4.488161122946569, −3.949102163565777, −3.433980611253844, −2.754220093428749, −2.136700611900304, −1.390596004696884, −0.7851656951608406, 0,
0.7851656951608406, 1.390596004696884, 2.136700611900304, 2.754220093428749, 3.433980611253844, 3.949102163565777, 4.488161122946569, 5.112014812099292, 5.411803557713758, 6.366199601357197, 6.457457790887647, 6.841669678766806, 7.584408446460277, 8.097040886271201, 8.572456840627392, 9.159638914106596, 9.629883511980523, 9.979621518450284, 10.61793474518439, 10.95577369552714, 11.52267206998452, 11.96599423499790, 12.22543511369940, 13.00553316883597, 13.13468886593905