L(s) = 1 | + 3-s − 3·7-s + 9-s + 5·11-s + 3·13-s − 7·19-s − 3·21-s − 4·23-s + 27-s + 29-s + 6·31-s + 5·33-s + 6·37-s + 3·39-s + 43-s − 3·47-s + 2·49-s − 12·53-s − 7·57-s + 4·59-s + 12·61-s − 3·63-s + 10·67-s − 4·69-s − 8·71-s + 16·73-s − 15·77-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.13·7-s + 1/3·9-s + 1.50·11-s + 0.832·13-s − 1.60·19-s − 0.654·21-s − 0.834·23-s + 0.192·27-s + 0.185·29-s + 1.07·31-s + 0.870·33-s + 0.986·37-s + 0.480·39-s + 0.152·43-s − 0.437·47-s + 2/7·49-s − 1.64·53-s − 0.927·57-s + 0.520·59-s + 1.53·61-s − 0.377·63-s + 1.22·67-s − 0.481·69-s − 0.949·71-s + 1.87·73-s − 1.70·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.715598916\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.715598916\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 43 | \( 1 - T \) |
good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−14.52791736429971, −13.97717402188115, −13.46956781763163, −12.98625280384114, −12.51997232230350, −12.06532236567636, −11.34561778281204, −10.95265105066996, −10.15914349734909, −9.775992395684325, −9.283100089247861, −8.815963310928188, −8.181320039228330, −7.892152300098166, −6.678637375991654, −6.568524951774452, −6.297316282164912, −5.422567953838954, −4.431497373867461, −3.991593392133347, −3.612072823474743, −2.848238273872081, −2.175268858381489, −1.407628015470235, −0.5704524564065271,
0.5704524564065271, 1.407628015470235, 2.175268858381489, 2.848238273872081, 3.612072823474743, 3.991593392133347, 4.431497373867461, 5.422567953838954, 6.297316282164912, 6.568524951774452, 6.678637375991654, 7.892152300098166, 8.181320039228330, 8.815963310928188, 9.283100089247861, 9.775992395684325, 10.15914349734909, 10.95265105066996, 11.34561778281204, 12.06532236567636, 12.51997232230350, 12.98625280384114, 13.46956781763163, 13.97717402188115, 14.52791736429971