L(s) = 1 | − 2·7-s + 3·11-s + 2·13-s − 3·17-s + 19-s + 6·23-s − 5·25-s + 6·29-s + 4·31-s − 4·37-s + 9·41-s + 43-s + 6·47-s − 3·49-s + 12·53-s − 3·59-s + 8·61-s − 5·67-s + 12·71-s + 11·73-s − 6·77-s + 4·79-s − 12·83-s + 6·89-s − 4·91-s + 5·97-s − 14·103-s + ⋯ |
L(s) = 1 | − 0.755·7-s + 0.904·11-s + 0.554·13-s − 0.727·17-s + 0.229·19-s + 1.25·23-s − 25-s + 1.11·29-s + 0.718·31-s − 0.657·37-s + 1.40·41-s + 0.152·43-s + 0.875·47-s − 3/7·49-s + 1.64·53-s − 0.390·59-s + 1.02·61-s − 0.610·67-s + 1.42·71-s + 1.28·73-s − 0.683·77-s + 0.450·79-s − 1.31·83-s + 0.635·89-s − 0.419·91-s + 0.507·97-s − 1.37·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.616972048\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.616972048\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−9.515144727195132739534503821127, −8.982636192572195812922115743561, −8.131333319129826356153375895616, −6.97337389606279696915121240099, −6.48609808834188223043361623015, −5.57014449737897797729246275255, −4.39002474802971059462229776672, −3.57376731663222433353058317817, −2.48468545989121014995168597575, −0.967496248641159004586773303229,
0.967496248641159004586773303229, 2.48468545989121014995168597575, 3.57376731663222433353058317817, 4.39002474802971059462229776672, 5.57014449737897797729246275255, 6.48609808834188223043361623015, 6.97337389606279696915121240099, 8.131333319129826356153375895616, 8.982636192572195812922115743561, 9.515144727195132739534503821127