L(s) = 1 | − 3-s − 2·7-s + 9-s − 4·11-s − 13-s − 17-s + 8·19-s + 2·21-s − 27-s + 2·29-s + 6·31-s + 4·33-s + 4·37-s + 39-s + 2·41-s − 4·43-s − 6·47-s − 3·49-s + 51-s − 6·53-s − 8·57-s + 10·59-s − 10·61-s − 2·63-s + 4·67-s − 8·71-s + 8·77-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s − 1.20·11-s − 0.277·13-s − 0.242·17-s + 1.83·19-s + 0.436·21-s − 0.192·27-s + 0.371·29-s + 1.07·31-s + 0.696·33-s + 0.657·37-s + 0.160·39-s + 0.312·41-s − 0.609·43-s − 0.875·47-s − 3/7·49-s + 0.140·51-s − 0.824·53-s − 1.05·57-s + 1.30·59-s − 1.28·61-s − 0.251·63-s + 0.488·67-s − 0.949·71-s + 0.911·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 265200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 265200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−13.10102380590468, −12.48266157862505, −12.16188049234151, −11.56210211564442, −11.35946100562892, −10.63469197549600, −10.20078851830600, −9.942627744555140, −9.381809931040366, −9.050412839695097, −8.146890059472607, −7.882778859600338, −7.430404797780364, −6.856439020843058, −6.303996711156572, −6.035119546843011, −5.234315308480814, −5.010218703207252, −4.563669262516269, −3.702867175746536, −3.202281396032343, −2.769922595769844, −2.171156854010432, −1.295966585368103, −0.6822293684216237, 0,
0.6822293684216237, 1.295966585368103, 2.171156854010432, 2.769922595769844, 3.202281396032343, 3.702867175746536, 4.563669262516269, 5.010218703207252, 5.234315308480814, 6.035119546843011, 6.303996711156572, 6.856439020843058, 7.430404797780364, 7.882778859600338, 8.146890059472607, 9.050412839695097, 9.381809931040366, 9.942627744555140, 10.20078851830600, 10.63469197549600, 11.35946100562892, 11.56210211564442, 12.16188049234151, 12.48266157862505, 13.10102380590468