L(s) = 1 | − 5-s − 7-s − 2·11-s − 6·13-s − 6·19-s − 23-s + 25-s − 4·29-s + 2·31-s + 35-s − 2·37-s − 2·41-s + 10·43-s + 6·47-s + 49-s − 6·53-s + 2·55-s + 4·59-s − 6·61-s + 6·65-s − 6·67-s − 2·71-s + 6·73-s + 2·77-s + 8·79-s − 14·83-s + 2·89-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.377·7-s − 0.603·11-s − 1.66·13-s − 1.37·19-s − 0.208·23-s + 1/5·25-s − 0.742·29-s + 0.359·31-s + 0.169·35-s − 0.328·37-s − 0.312·41-s + 1.52·43-s + 0.875·47-s + 1/7·49-s − 0.824·53-s + 0.269·55-s + 0.520·59-s − 0.768·61-s + 0.744·65-s − 0.733·67-s − 0.237·71-s + 0.702·73-s + 0.227·77-s + 0.900·79-s − 1.53·83-s + 0.211·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.94123584540536, −13.22618057974131, −12.74637587565530, −12.47030519288376, −12.03111426538398, −11.45869965769029, −10.75269911385244, −10.59637390219539, −9.845882345728363, −9.594357832183166, −8.850417873188275, −8.492839738827155, −7.762194487066159, −7.408389312910276, −7.037657155721928, −6.254014297233636, −5.875219776303366, −5.119678502927463, −4.673712791831653, −4.159529115594760, −3.546814121405125, −2.758756598354821, −2.396720438679741, −1.739020356431464, −0.5896162895501723, 0,
0.5896162895501723, 1.739020356431464, 2.396720438679741, 2.758756598354821, 3.546814121405125, 4.159529115594760, 4.673712791831653, 5.119678502927463, 5.875219776303366, 6.254014297233636, 7.037657155721928, 7.408389312910276, 7.762194487066159, 8.492839738827155, 8.850417873188275, 9.594357832183166, 9.845882345728363, 10.59637390219539, 10.75269911385244, 11.45869965769029, 12.03111426538398, 12.47030519288376, 12.74637587565530, 13.22618057974131, 13.94123584540536