L(s) = 1 | − 3-s + 2·7-s − 2·9-s + 3·11-s − 6·13-s − 3·17-s + 5·19-s − 2·21-s − 4·23-s + 5·27-s + 4·29-s + 2·31-s − 3·33-s − 4·37-s + 6·39-s + 9·41-s − 4·43-s − 3·49-s + 3·51-s − 2·53-s − 5·57-s + 4·59-s − 61-s − 4·63-s − 5·67-s + 4·69-s + 14·71-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.755·7-s − 2/3·9-s + 0.904·11-s − 1.66·13-s − 0.727·17-s + 1.14·19-s − 0.436·21-s − 0.834·23-s + 0.962·27-s + 0.742·29-s + 0.359·31-s − 0.522·33-s − 0.657·37-s + 0.960·39-s + 1.40·41-s − 0.609·43-s − 3/7·49-s + 0.420·51-s − 0.274·53-s − 0.662·57-s + 0.520·59-s − 0.128·61-s − 0.503·63-s − 0.610·67-s + 0.481·69-s + 1.66·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 61 | \( 1 + T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 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
−15.83565339740019, −14.96098107041864, −14.55793379386590, −14.11281742169935, −13.77003510145359, −12.81385528456436, −12.22123194865288, −11.81755500507256, −11.49207789325161, −10.92731841582742, −10.16134891053845, −9.723098660039946, −9.072322458691038, −8.473765073592847, −7.866830314581279, −7.233241581643954, −6.695685218937822, −5.987239704043112, −5.419490539823436, −4.725700466919276, −4.429159577694405, −3.381780379204103, −2.632865743249601, −1.919606279294984, −0.9707842005996689, 0,
0.9707842005996689, 1.919606279294984, 2.632865743249601, 3.381780379204103, 4.429159577694405, 4.725700466919276, 5.419490539823436, 5.987239704043112, 6.695685218937822, 7.233241581643954, 7.866830314581279, 8.473765073592847, 9.072322458691038, 9.723098660039946, 10.16134891053845, 10.92731841582742, 11.49207789325161, 11.81755500507256, 12.22123194865288, 12.81385528456436, 13.77003510145359, 14.11281742169935, 14.55793379386590, 14.96098107041864, 15.83565339740019